mathematics-class-12-unit-02-chapter-03-inverse-trigonometric-functions-l-3-5-slcxfosdc8u

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- <ins>**Problem-1**</ins>

- Find <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </math> such that
- <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>cos</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>.</mo> </mstyle> </math>

- <ins>**Solution:**</ins>

- sin(cos$^{-1}$(x) = $\frac{1}{2}$ = sin($\frac{\pi}{6}$)

- $x=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{6}$

- But, x must be negative.
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<footer style = "font-size: 18px"> $\rightarrow$ Inverse trigonometric functions (ITF) lecture 3 $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- If -1 < x < 0 , then

- cos$^{-1}x \in ({\frac{\pi}{2}},{\pi})$ But sin$^{-1} ({\frac{1}{2}})$ $\in ({\frac{-\pi}{2},\frac{\pi}{2}}) \leftarrow$ Range set of sin$^{-1}$
- Find 
- <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </math>

- such that

- $\sin\left(\cos^{-1}x\right)=\frac{1}{2}=\sin\frac{\pi}{6}$

- <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <msup> <mi>cos</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>=</mo> <mrow> <mo></mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo></mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </math>

- $x=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{6}$
  
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<footer style = "font-size: 18px">Inverse trigonometric functions (ITF) lecture 3 $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- Using supplementary angles

- For $-1<x<0, \theta=\cos ^{-1} x \in\left(\frac{\pi}{2}, \pi\right)$

- But, $\sin (\pi-\theta)=\sin \theta={\sin \left(\cos ^{-1} x\right)}$

- $\pi-\theta \in(0, \frac{\pi} { 2}) \in $ Range set of sin$^{-1}$

- $\sin (\pi-\theta)=\frac{1} {2} . $

- $\Rightarrow  \pi-\theta=\sin ^{-1} (\frac{1} {2}) $

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### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Trigonometric functions  and their inverses </primary>
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<div style='float: left; width:99%;font-size:30px;'>

- $f$ : Trigonometric functions $(\sin , \cos , \tan , \cot$, sec,  cosec)

- $f: A \longrightarrow B$

- $f^{-1}: B \longrightarrow C⊂A$

- If $f(\theta)=x, \theta \in C$ then $\theta=f^{-1}(x)$

- If $f(\theta)=x, \theta \notin C$ then $\theta=$ ? $\theta \neq f^{-1}(x)$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Trigonometric functions  and their inverses </span> $\rightarrow$ Sine function and it's inverse  $\rightarrow$ Sine function and it's inverse </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-1"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-sine-function-and-its-inverse-primary"><primary style="font-size:24px"> Sine function and it’s inverse </primary></h3>
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<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>$\sin : \mathbb{R} \longrightarrow[-1,1]$</p>
</li>
<li>
<p>$\sin ^{-1}:[-1,1] \longrightarrow[\frac{-\pi} {2}, \frac{\pi} {2}]$</p>
</li>
<li>
<p>$\sin (\theta)=x, x \in[-1,1]$</p>
</li>
<li>
<p>Find $ \theta, \text { if } \theta \in\left[m \pi-\frac{\pi}{2}, {m \pi +\frac{\pi}{2}}\right]$ , where m is an integer.</p>
</li>
<li>
<p>$\sin (\theta)=x,$</p>
</li>
<li>
<p>$m=0, \theta \in [\frac{-\pi} {2}, \frac{\pi} {2}]$</p>
</li>
<li>
<p>$ \theta=\sin ^{-1} x$</p>
</li>
</ul>
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<footer style = "font-size: 18px"> Solution $\rightarrow$ Trigonometric functions  and their inverses $\rightarrow$ <span style = "color:blue"> Sine function and it's inverse </span> $\rightarrow$ Sine function and it's inverse $\rightarrow$ Sine function and it's inverse </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-2"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-sine-function-and-its-inverse-primary-1"><primary style="font-size:24px"> Sine function and it’s inverse </primary></h3>
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<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>$ \sin \theta=x, \quad \theta \in\left[m \pi-\frac{\pi}{2}, m \pi-\frac{\pi}{2}\right]$</p>
</li>
<li>
<p>$ \theta-m \pi \in [-\frac{\pi}{2}, \frac{\pi}{2}]$  $\leftarrow$ Range set of sin$^{-1}$</p>
</li>
<li>
<p>$ \sin (\theta-m \pi)=\sin \theta \cos m \pi$</p>
</li>
<li>
<p>$ =(-1)^m$</p>
</li>
<li>
<p>$ =(-1)^m \sin \theta$</p>
</li>
<li>
<p>$ =(-1)^m x $</p>
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</ul>
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<hr>
<footer style = "font-size: 18px">Trigonometric functions  and their inverses $\rightarrow$ Sine function and it's inverse $\rightarrow$ <span style = "color:blue"> Sine function and it's inverse </span> $\rightarrow$ Sine function and it's inverse $\rightarrow$ Sine function and it's inverse </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Sine function and it's inverse </primary>
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- **m  odd ;**

- $\sin (\theta-m \pi)=(-1)^m x =-x$

- $\theta-m \pi=\sin ^{-1}(-x) =-\sin ^{-1} x$

- $\theta=m \pi-\sin ^{-1} x{. }$

- **In general**,

- $\theta \in\left[m \pi-\frac{\pi}{2}, m \pi+ \frac{\pi}{2}\right]$  s.t. $\sin \theta=x,|x| \leqslant 1$

- $\theta = m \pi + \sin ^{-1} x , $ m is even

- $\theta = m \pi - \sin ^{-1} x , $ m is odd

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<hr>
<footer style = "font-size: 18px">Trigonometric functions  and their inverses $\rightarrow$ Sine function and it's inverse $\rightarrow$ <span style = "color:blue"> Sine function and it's inverse </span> $\rightarrow$ Addition of tangent inverse functions $\rightarrow$ Addition of tangent inverse functions </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-3"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-addition-of-tangent-inverse-functions-primary"><primary style="font-size:24px"> Addition of tangent inverse functions </primary></h3>
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<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p><ins><strong>Show :</strong></ins></p>
</li>
<li>
<p>$ \tan \left(\tan ^{-1} x+\tan ^{-1} y\right)=\frac{\tan \left(\tan ^{-1} x\right)+\tan \left(\tan ^{-1} y\right)}{1-\tan \left(\tan ^{-1} x\right) \tan \left(\tan ^{-1} y\right)}=\frac{x+y}{1-x y}$</p>
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<li>
<p><ins><strong>Solution :</strong></ins></p>
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<li>
<p>Let,</p>
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<p>$ A=\tan ^{-1} x, ~ ~ B=\tan ^{-1} y$</p>
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<li>
<p>$ \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$</p>
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</ul>
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<hr>
<footer style = "font-size: 18px">Sine function and it's inverse $\rightarrow$ Sine function and it's inverse $\rightarrow$ <span style = "color:blue"> Addition of tangent inverse functions</span> $\rightarrow$ Addition of tangent inverse functions $\rightarrow$ Conditions for $\tan ^{-1} x + \tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-4"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-addition-of-tangent-inverse-functions-primary-1"><primary style="font-size:24px"> Addition of tangent inverse functions </primary></h3>
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<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>$\tan \left(\tan ^{-1} x+\tan ^{-1} y\right)= \frac{x+y}{1-x y}$</p>
</li>
<li>
<p>$ \tan ^{-1} x+\tan ^{-1} y = ? $</p>
</li>
<li>
<p><strong>In general,</strong></p>
</li>
<li>
<p>$ \tan ^{-1} x+\tan ^{-1} y  \neq \tan ^{-1}\left(\frac{x+y}{1-x y}\right)$</p>
</li>
<li>
<p>$\because\quad$ Range set for tan$^{-1}$ is $(\frac{-\pi}{2},\frac{\pi}{2})$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Sine function and it's inverse $\rightarrow$ Addition of tangent inverse functions $\rightarrow$ <span style = "color:blue"> Addition of tangent inverse functions </span> $\rightarrow$ Conditions for $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-5"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-conditions-for-tan--1-x--tan--1-y-primary"><primary style="font-size:24px"> Conditions for $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
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<table>
<thead>
<tr>
<th><strong>Condition</strong></th>
<th>$\tan ^{-1} x + \tan ^{-1} y$</th>
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<td>$ xy < 1 $</td>
<td>$ \qquad  \tan ^{-1} (\frac{x+y}{1-xy})$</td>
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<tr>
<td>$ x > 0,y > 0 ,xy>1 $</td>
<td>$ \quad  \pi + \tan ^{-1} (\frac{x+y}{1-xy}) $</td>
</tr>
<tr>
<td>$ x < 0, y < 0, xy > 1 $</td>
<td>$    -\pi + \tan ^{-1} (\frac{x+y}{1-xy}) $</td>
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</table>
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<hr>
<footer style = "font-size: 18px"> Addition of tangent inverse functions $\rightarrow$ Addition of tangent inverse functions $\rightarrow$ <span style = "color:blue">Conditions for $\tan ^{-1} x + \tan ^{-1} y$</span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary>
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- We know,

- $\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, x \in \mathbb{R} $

- $\tan ^{-1}\left(\frac{1}{x}\right)=\cot ^{-1} x, x>0$

- Add the above formulas we get,

- $\tan ^{-1} x+\tan ^{-1}\left(\frac{1}{x}\right)=\frac{\pi}{2}, x>0$

- $ \Rightarrow   -\tan ^{-1} x-\tan ^{-1}\left(\frac{1}{x}\right)=\frac{-\pi}{2},x>0$

- $\tan ^{-1}(-x)+\tan ^{-1}(\frac{1}{-x})=\frac{-\pi}{2},-x<0$  
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<hr>
<footer style = "font-size: 18px"> Addition of tangent inverse functions $\rightarrow$ Conditions for $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-6"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-proof-for--conditions-of-tan--1-x--tan--1-y-primary"><primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
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<div style='float: left; width:60%;font-size:28px;'>
<ul>
<li>
<p>Put,</p>
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<li>
<p>$y=-x$</p>
</li>
<li>
<p>$\tan ^{-1} y+\tan ^{-1} (\frac{1}{y})=\frac{-\pi}{2}, y<0$</p>
</li>
<li>
<p><ins><strong>Condition-1:</strong></ins></p>
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<li>
<p>For xy < 1</p>
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<li>
<p><ins><strong>Case (i)</strong></ins></p>
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<li>
<p>$ x>0, y>0, x y<1$</p>
</li>
<li>
<p>$\tan ^{-1} x \in\left(0, \frac{\pi}{2}\right) , \tan ^{-1} y \in\left(0, \frac{\pi}{2}\right)$</p>
</li>
<li>
<p>$ 0 < y < \frac{1}{x}$</p>
</li>
</ul>
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<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/61b4fa9c-6480-4d8e-2738-a4f969cd5a00/public" alt="">

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<hr>
<footer style = "font-size: 18px"> Conditions for $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-7"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-proof-for--conditions-of-tan--1-x--tan--1-y-primary-1"><primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
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<div style='float: left; width:70%;font-size:23px;'>
<ul>
<li>
<p>$0<\tan ^{-1} y<\tan ^{-1} (\frac{1}{x})…(a)$</p>
</li>
<li>
<p>We know, $ 0<\tan^{-1}x$ (See graph) and add $ \tan ^{-1} x $ to equation (a)</p>
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<li>
<p>$\Rightarrow 0<\tan ^{-1} x<\tan ^{-1} x+\tan ^{-1} y <\tan ^{-1} x+\tan ^{-1} (\frac{1}{x})=\frac{\pi}{2}$</p>
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</ul>
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<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$  </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-8"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-proof-for--conditions-of-tan--1-x--tan--1-y-primary-2"><primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
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<div style='float: left; width:70%;font-size:23px;'>
<ul>
<li>
<p><ins><strong>Condition-1:</strong></ins> For xy < 1</p>
</li>
<li>
<p><ins><strong>Case (ii)</strong></ins></p>
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<li>
<p>$x<0, y<0, x y<1$</p>
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<li>
<p>$\tan ^{-1} x \in\left(\frac{-\pi}{2}, 0\right) , \tan ^{-1} y \in\left(\frac{-\pi}{2}, 0\right)$</p>
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<li>
<p>$y>\frac{1}{x}$</p>
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<li>
<p>$0>\tan ^{-1} y>\tan ^{-1} (\frac{1}{x})…(b)$</p>
</li>
<li>
<p>We know, $ 0>\tan^{-1}x$ (See graph) and add $ \tan ^{-1} x $ to equation (b)</p>
</li>
<li>
<p>$ \Rightarrow 0>\tan ^{-1} x>\tan ^{-1} x+\tan ^{-1} y>\tan ^{-1} x +\tan ^{-1} \frac{1}{x}$</p>
</li>
<li>
<p>$(\because\quad \tan ^{-1} x +\tan ^{-1} \frac{1}{x}=\frac{-\pi}{2})$</p>
</li>
<li>
<p>$ \Rightarrow 0>\tan ^{-1} x+\tan ^{-1} y>\frac{-\pi}{2}$</p>
</li>
</ul>
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<div style='float: right; width:30%;'>
<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/98e58805-3af8-4f7d-fbc1-da05b2d2d600/public" alt="">

</div>
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<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-9"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-proof-for--conditions-of-tan--1-x--tan--1-y-primary-3"><primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
<hr>
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<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p><ins><strong>Case (iii)</strong></ins></p>
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<li>
<p>For xy $\leq$ 0</p>
</li>
<li>
<p>$x y\leqslant 0 $</p>
</li>
<li>
<p>$ x \leqslant 0, y \geqslant 0$</p>
</li>
<li>
<p>$\tan ^{-1} x \in[\frac{-\pi}{2},0], \tan ^{-1} y \in\left[0, \frac{\pi}{2}\right]$</p>
</li>
<li>
<p>$\Rightarrow\tan ^{-1} x+\tan ^{-1} y \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$</p>
</li>
<li>
<p>$\quad x \geqslant 0, y \leqslant 0$</p>
</li>
<li>
<p>$\tan ^{-1} x \in\left[0, \frac{\pi}{2}\right], \tan ^{-1} y \in\left[\frac{-\pi}{2}, 0\right]$</p>
</li>
<li>
<p>$\tan ^{-1} x+\tan ^{-1} y \in[\frac{-\pi}{2} , \frac{\pi}{2}]$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-10"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-proof-for--conditions-of-tan--1-x--tan--1-y-primary-4"><primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
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<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p><ins><strong>Condition-2:</strong></ins></p>
</li>
<li>
<p>$ ~ ~ x>0, y>0, \quad x y>1$</p>
</li>
<li>
<p>$\tan ^{-1} x \in (0, \frac{\pi}{2} )$</p>
</li>
<li>
<p>$\tan ^{-1} y \in(0, \frac{\pi}{2} )$</p>
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<li>
<p>$y> \frac{1}{x}$</p>
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<li>
<p>$ \Rightarrow \tan ^{-1} y>\tan ^{-1} (\frac{1}{x})$</p>
</li>
<li>
<p>$\tan ^{-1} x+\tan ^{-1} y>\underbrace{\tan ^{-1} x+\tan ^{-1} \frac{1}{x}}_{=\frac{\pi}{2}}$</p>
</li>
<li>
<p>$\Rightarrow \tan ^{-1} x+\tan ^{-1} y> \frac{\pi}{2}$</p>
</li>
</ul>
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</main>
<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Formula for $2 \tan^{-1} x$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-11"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-proof-for--conditions-of-tan--1-x--tan--1-y-primary-5"><primary style="font-size:24px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p><ins><strong>Condition-2:</strong></ins></p>
</li>
<li>
<p>$\theta=\tan ^{-1} x+\tan ^{-1} y>\frac{\pi}{2}$</p>
</li>
<li>
<p>$\Rightarrow\theta<\pi ~ ~ ~  \because\tan^{-1} x < \frac{\pi}{2}$ & $\tan^{-1}y<\frac{\pi}{2}$</p>
</li>
<li>
<p>$\Rightarrow\left.\theta \in (\frac{\pi}{2},\pi\right)\theta ∉$ Range set of $f^{-1}$</p>
</li>
<li>
<p>But, $\left.\theta-\pi \in (\frac{-\pi}{2}, 0\right)$ ⊂ Range set of $\tan^{-1}$</p>
</li>
<li>
<p>$\Rightarrow\tan (\theta-\pi)=\tan \theta=\frac{x+y}{1-x y}$</p>
</li>
<li>
<p>$ \theta-\pi = \tan^{-1}\frac{x+y}{1-x y}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ </span> $\rightarrow$ Formula for $2 \tan^{-1} x$  $\rightarrow$ Formula for $\tan ^{-1} x-\tan ^{-1} y$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-12"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-formula-for-2-tan-1-x-primary"><primary style="font-size:24px"> Formula for $2 \tan^{-1} x$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>$\rightarrow$ For, $|x|≤ 1$</p>
</li>
<li>
<p>$ 2 \tan^{-1} x = \tan^{-1}[\frac{2x}{1-x^2}]$</p>
</li>
<li>
<p>$\rightarrow$ For, $ x > 1$</p>
</li>
<li>
<p>$ 2 \tan^{-1} x = \pi + \tan^{-1}[\frac{2x}{1-x^2}]$</p>
</li>
<li>
<p>$\rightarrow$ For, $ x<-1$</p>
</li>
<li>
<p>$ 2 \tan^{-1} x = - \pi + \tan^{-1}[\frac{2x}{1-x^2}]$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Formula for $2 \tan^{-1} x$ </span> $\rightarrow$ Formula for $\tan ^{-1} x-\tan ^{-1} y$ $\rightarrow$ Conversion from $ \sin^{-1} \text{and} \tan^{-1}$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-13"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-formula-for-tan--1-x-tan--1-y-primary"><primary style="font-size:24px"> Formula for $\tan ^{-1} x-\tan ^{-1} y$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:20px;'>
<ul>
<li>
<p>$\because \tan^{-1}$ is odd function</p>
</li>
<li>
<p>$\Rightarrow\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}(-y)$</p>
</li>
</ul>
<table>
<thead>
<tr>
<th>Conditions</th>
<th>$ {\tan ^{-1} x-\tan ^{-1} y}$</th>
</tr>
</thead>
<tbody>
<tr>
<td>$x (-y) < 1$ ,$xy>-1$</td>
<td>$ \tan^{-1} (\frac{x+(-y)}{1-x(-y)})$ =  $ \tan^{-1}(\frac{x-y}{1+xy})$</td>
</tr>
<tr>
<td>$x>0,-y>0,x(-y)>1$ $x>0, y<0, xy<-1$</td>
<td>$ \pi+ \tan^{-1}(\frac{x+(-y)}{1-x(-y)})$ = $\quad \quad \quad $  $ \pi+ \tan^{-1}(\frac{x-y}{1+xy})$</td>
</tr>
<tr>
<td>$x<0,-y<0,x(-y)>1$$x<0, y>0, xy<-1$</td>
<td>$ - \pi+ \tan^{-1}(\frac{x+(-y)}{1-x(-y)})$ = $\quad  \quad $ $ - \pi+ \tan^{-1}(\frac{x-y}{1+xy})$</td>
</tr>
</tbody>
</table>
</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px"> Proof for  conditions of $\tan ^{-1} x + \tan ^{-1} y$ $\rightarrow$ Formula for $2 \tan^{-1} x$ $\rightarrow$ <span style = "color:blue"> Formula for $\tan ^{-1} x-\tan ^{-1} y$ </span> $\rightarrow$ Conversion from $ \sin^{-1} \text{and} \tan^{-1}$ $\rightarrow$ Reverse conversion from $\tan^{-1} \text{to} \sin^{-1}$ </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-14"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-conversion-from--sin-1-textand-tan-1-primary"><primary style="font-size:24px"> Conversion from $ \sin^{-1} \text{and} \tan^{-1}$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>To make identity something like, $\sin ^{-1} x=\tan ^{-1}(?) , |x| \leqslant 1$</p>
</li>
<li>
<p>Let,$ ~ \theta=\sin ^{-1} x,\theta \in[-\frac{\pi}{2}, \frac{\pi}{2}]$</p>
</li>
<li>
<p>$\tan \theta= \frac{\sin \theta }{\cos \theta}= \frac{x}{\sqrt{1-x^2} \quad}$</p>
</li>
<li>
<p>$\because\cos ^2 \theta =1-\sin ^2 \theta =1-x^2 \Rightarrow \cos \theta= \sqrt{1-x^2}~  $because $\cos \theta>0$</p>
</li>
<li>
<p>$\tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^2}} $</p>
</li>
<li>
<p>$\Rightarrow\theta=\tan^{-1} (\frac{x}{\sqrt{1-x^2}}) \quad \because  \theta = \sin^{-1}x$</p>
</li>
<li>
<p>$\Rightarrow\sin^{-1}x = \tan^{-1}\frac{x}{\sqrt{1-x^2}}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>


</div>
</main>
<hr>
<footer style = "font-size: 18px"> Formula for $2 \tan^{-1} x$ $\rightarrow$ Formula for $\tan ^{-1} x-\tan ^{-1} y$ $\rightarrow$ <span style = "color:blue"> Conversion from $ \sin^{-1} \text{and} \tan^{-1}$ </span> $\rightarrow$ Reverse conversion from $\tan^{-1} \text{to} \sin^{-1}$ $\rightarrow$ Problem  </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-15"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-reverse-conversion-from-tan-1-textto-sin-1-primary"><primary style="font-size:24px"> Reverse conversion from $\tan^{-1} \text{to} \sin^{-1}$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>In, $~ ~\tan ^{-1}(x)=\sin ^{-1}(?), x \in \mathbb{R}$  we have to find out(?)</p>
</li>
<li>
<p>Let$ ~ \theta$, $\theta=\tan ^{-1}(x) \in( \frac{-\pi}{2} , \frac{\pi}{2})$</p>
</li>
<li>
<p>$\sin \theta=\frac{\tan \theta}{\operatorname{sec} \theta}=\frac{\tan \theta}{\sqrt{1+\tan ^2 \theta}} \because \sec \theta>0$</p>
</li>
<li>
<p>$ = \frac{x}{\sqrt{1+x^2}}$</p>
</li>
<li>
<p>$\because \ x = \tan\theta 1+ \tan^{2} \theta= \sec^{2} \theta$</p>
</li>
<li>
<p>$\Rightarrow\sin \theta=\frac{x}{\sqrt{1+x^2}} \quad = \sin ({\tan ^{-1} x}) = \frac{x}{\sqrt{1+x^2}}$</p>
</li>
<li>
<p>$\Rightarrow\tan ^{-1} x = \sin^{-1}  \frac{x} {\sqrt{1+x^2}} $</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px"> Formula for $\tan ^{-1} x-\tan ^{-1} y$ $\rightarrow$ Conversion from $ \sin^{-1} \text{and} \tan^{-1}$ $\rightarrow$ <span style = "color:blue"> Reverse conversion from $\tan^{-1} \text{to} \sin^{-1}$ </span> $\rightarrow$ Problem $\rightarrow$ Solution  </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-16"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-problem-primary"><primary style="font-size:24px"> Problem </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p><strong><ins>Problem-2</ins></strong></p>
</li>
<li>
<p>Find</p>
</li>
<li>
<p>$\sin ^{-1}\left(\frac{1}{3}\right)+\sin^{-1}\left[\frac{2}{3}\left(1-\frac{1}{\sqrt{8}}\right)\right]=?$</p>
</li>
<li>
<p><strong><ins>Solution</ins></strong></p>
</li>
<li>
<p>$\sin ^{-1}(x)=\tan ^{-1} \frac{x}{\sqrt{1-x^2}}$</p>
</li>
<li>
<p>$\sin ^{-1}\left(\frac{1}{3}\right)=\tan ^{-1}\left(\frac{\frac{1}{3}}{\sqrt{\frac{1-1}{9}}}\right)=\tan ^{-1}\left(\frac{1}{\sqrt{8}}\right)$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px"> Conversion from $ \sin^{-1} \text{and} \tan^{-1}$ $\rightarrow$ Reverse conversion from $\tan^{-1} \text{to} \sin^{-1}$ $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Solution $\rightarrow$  Solution </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-3-success-17"><Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-solution-primary"><primary style="font-size:24px"> Solution </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px;'>
<ul>
<li>
<p>$\sin ^{-1}\left(\frac{2}{3}\left(1-\frac{1}{\sqrt{8}}\right)\right)=\tan ^{-1}\left[\frac{\frac{2}{3}(\frac{1-1}{ \sqrt{8}})}{\left.\sqrt{1-\frac{4}{9}(1 \frac{-1}{\sqrt{8}}}^2\right)}\right]$</p>
</li>
<li>
<p>$=\tan ^{-1} \frac{2(\frac{1-1}{\sqrt{8}})}{\left.\sqrt{9-4\left(1+\frac{1}{8}-\frac{2}{\sqrt{8}}\right.}\right)}$</p>
</li>
<li>
<p>$\tan ^{-1} \frac{2(1-\frac{1}{\sqrt{8}})}{\sqrt{5-\frac{1}{2}+\frac{8}{\sqrt{8}}}}=\tan ^{-1} \frac{2(1-\frac{1}{\sqrt{8}})}{\sqrt{\frac{9}{2}+\sqrt{8}}}$</p>
</li>
<li>
<p>$ \tan ^{-1}\frac{2(\sqrt{8}-1)}{\sqrt{36+8 \sqrt{8}}}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Reverse conversion from $\tan^{-1} \text{to} \sin^{-1}$ $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- $\sin ^{-1} \frac{1}{3}+\sin ^{-1}\left[\frac{2}{3}\left(1-\frac{1}{\sqrt{8}}\right)\right]$

- $=\tan ^{-1}\left(\frac{1}{\sqrt{8}}\right)+\tan ^{-1}\left(\frac{(\sqrt{8}-1)}{\sqrt{9+2 \sqrt{8}}}\right)$

- $ \tan ^{-1}x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}, x y<1 $

- $ =\tan ^{-1}\left(\frac{1}{\sqrt{8}}+\frac{(\sqrt{8}-1)}{\sqrt{9+2 \sqrt{8}}}\right)=\tan ^{-1}(1)$

- $\left(1 -\frac{(\sqrt{8-1)}}{\sqrt{72+16 \sqrt{8}}}\right)=\frac{\pi}{4}$

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</main>
<hr>
<footer style = "font-size: 18px">Problem $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Thank you $\rightarrow$  </footer>