mathematics-class-12-unit-02-chapter-02-inverse-trigonometric-functions-l-2-5-x2ym2knzdc0

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Recap </primary>
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|Function | Domain | Range|
|---|---|---|
| <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo></mo> </mrow> </mstyle> </math>   | [-1, 1]      |<math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mfrac> <mrow> <mo>-</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>]</mo> </mrow> </mstyle> </math> |
|<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> <mo></mo> </mrow> </mstyle> </math>  | [-1, 1]     | [0,π]| 
| <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <msup> <mi>tan</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo></mo> </mrow> </mstyle> </math>   | <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>ℝ</mo> </mstyle> </math>     |<math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>-</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>)</mo> </mrow> </mstyle> </math> |
| <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <msup> <mi>cot</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo></mo> </mrow> </mstyle> </math>   | <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>ℝ</mo> </mstyle> </math>     | (0,π) |   
| <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <msup> <mi>sec</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo></mo> </mrow> </mstyle> </math>    | <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>ℝ</mo> <mo>-</mo> <mrow> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mstyle> </math>|<math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo>]</mo> </mrow> <mo>-</mo> <mrow> <mo>{</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>}</mo> </mrow> </mstyle> </math>|
| <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msup> <mrow> <mtext>cosec</mtext> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mstyle> </math>    | <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>ℝ</mo> <mo>-</mo> <mrow> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mstyle> </math>     | <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mfrac> <mrow> <mo>-</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>]</mo> </mrow> <mo>-</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>}</mo> </mrow> </mstyle> </math>

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images//subject-images/iitpal/image/mathematics-class-12-unit-02-chapter-02-inverse-trigonometric-functions-l-2_5-x2ym2knzdc0-015-0064.8.jpg)

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<footer style = "font-size: 18px"> $\rightarrow$ Inverse trigonometric functions(ITF) lecture 2 $\rightarrow$ <span style = "color:blue"> Recap </span> $\rightarrow$ Caution $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-2-success-1"><Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-caution-primary"><primary style="font-size:24px"> Caution </primary></h3>
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<ul>
<li>
<p>$\sin ^{-1} x  \neq(\sin x)^{-1}$</p>
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<li>
<p>If $x=0$</p>
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<li>
<p>$\sin ^{-1} 0  =0 $</p>
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<li>
<p>$ \Rightarrow (\sin x)^{-1}  =(\sin 0)^{-1} $</p>
</li>
<li>
<p>$\Rightarrow \frac{1}{\sin 0}=\frac{1}{0}$</p>
</li>
<li>
<p>But, $\frac{1}{0}$ is not well-defined.</p>
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</ul>
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<footer style = "font-size: 18px">Inverse trigonometric functions(ITF) lecture 2 $\rightarrow$ Recap $\rightarrow$ <span style = "color:blue"> Caution </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Example </primary>
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- <ins>Example-1:</ins>

- Find the principal value of
- $\tan ^{-1}(-\sqrt{3}); $

- <b><ins>Solution:</ins></b>

- $\tan ^{-1}: R \rightarrow(-\frac{\pi}{2}, \frac{\pi}{2}) $  
- $ \tan \left(-\frac{\pi}{3}\right)=-\sqrt{3} $  
- $ \tan ^{-1}(-\sqrt{3})=-\frac{\pi}{3} $  
- $\tan \left(\pi-\frac{\pi}{3}\right)=-\sqrt{3}$

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<footer style = "font-size: 18px">Recap $\rightarrow$ Caution $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Solution </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-2-success-2"><Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-primary"><primary style="font-size:24px"> Example </primary></h3>
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<div style='float: left; width:99%; font-size:30px; margin-top:50px'>
<ul>
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<p><ins>Example-2:</ins></p>
</li>
<li>
<p>$\quad \operatorname{sin}: \mathbb{R} \rightarrow[-1,1]$</p>
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<li>
<p>$\sin ^{-1}:[-1,1] \rightarrow\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$</p>
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<p>For any $x \in[-1,1]$,</p>
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<li>
<p>$ {\sin \left(\sin ^{-1} x\right)= ? } $</p>
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<li>
<p><b><ins>Solution:</ins></b></p>
</li>
<li>
<p>$Let, \sin ^{-1} x=\theta \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] $</p>
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<li>
<p>$ \Rightarrow \sin (\sin ^{-1} x)=\sin\theta = x $</p>
</li>
<li>
<p>$  \Rightarrow \sin\theta = x $</p>
</li>
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<footer style = "font-size: 18px">Caution $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- <b><ins>Solution:</ins></b>

- $\sin ^{-1}(\sin y) \stackrel{?}{=} y $

- If $y=\frac{\pi}{6}, \sin \frac{\pi}{6}=\frac{1}{2} $

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

- $y=\frac{2 \pi}{3}, \sin \frac{2 \pi}{3}=\frac{\sqrt{3}}{2} $

- But, $ \sin ^{-1}(\sin y)=\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi}{3} . $

- $\because \sin ^{-1}(\sin y) \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \leftarrow $ 
- Range set of <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo></mo> </mrow> </mrow> </mstyle> </math>

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Important remark </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Solution </primary>
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<div style='float: left; width:99%; font-size:30px'>
	
- <b><ins>Solution:</ins></b>	
	
- $ y \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$    
- $ \sin ^{-1}(\sin y)=y$    
- $ \Rightarrow \sin ^{-1}(\sin y)=x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$   
- $ \sin \left(\sin ^{-1}(\underbrace{sin y}_{z})\right)=\sin x$

- Let , sin y = z  
- $ [\because \sin(\sin ^{-1} z)=z,|z| \leqslant 1 ]$   
- $\Rightarrow$ z = sin x

- sin y= sin x

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<footer style = "font-size: 18px">Example $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Important remark $\rightarrow$ Inverse trigonometric funtions & their range </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Important remark </primary>
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- If $ x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$    
- and
- $ y \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$

- also,	sin x = sin y

- Then, x = y

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<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/19c6a47c-df91-43ac-4211-52d464928300/public" >

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-12-unit-02-chapter-02-inverse-trigonometric-functions-l-2_5-x2ym2knzdc0-05.png) -->

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Important remark </span> $\rightarrow$ Inverse trigonometric funtions & their range $\rightarrow$ Inverse trigonometric funtions & their range </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Inverse trigonometric funtions & their range </primary>
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- $\text{ If y} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \text {,then, } $
 
- $\rightarrow \sin ^{-1}(\sin y)=y$

- $\rightarrow \cos \left(\cos ^{-1} x\right)=x, \quad|x| \leqslant 1$

- $\rightarrow \cos ^{-1}(\cos \theta)=\theta, \quad \theta \in[0, \pi]$

- $\rightarrow \tan \left(\tan ^{-1} x\right)=x,x \in \mathbb{R}$

- $\rightarrow \tan ^{-1}(\tan \theta)=\theta,\theta \in(-\pi / 2, \pi / 2)$      
- $\rightarrow \cot \left(\cot ^{-1} x\right)=x,  x \in \mathbb{R}$

- $\rightarrow \cot ^{-1}(\cot \theta)=\theta, \theta \in(0, \pi)$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Important remark $\rightarrow$ <span style = "color:blue"> Inverse trigonometric funtions & their range </span> $\rightarrow$ Inverse trigonometric funtions & their range $\rightarrow$ Relation between arcsin & arccosec funtions </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Inverse trigonometric funtions & their range </primary>
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- $\rightarrow \sec \left(\sec ^{-1} x\right)=x, \quad|x| \geqslant 1$

- $\rightarrow  \sec^{-1}(\sec \theta)=\theta,  \theta \in[0, \pi]-\\{\frac{\pi}{2}\\}$

- $\rightarrow  \operatorname{cosec}\left(\operatorname{cosec}^{-1} x\right)=x, \quad|x| \geqslant 1$

- $\rightarrow \operatorname{cosec}^{-1}(\operatorname{cosec} \theta)=\theta, \theta\in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\\{{0\\}}$

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<footer style = "font-size: 18px">Important remark $\rightarrow$ Inverse trigonometric funtions & their range $\rightarrow$ <span style = "color:blue"> Inverse trigonometric funtions & their range </span> $\rightarrow$ Relation between arcsin & arccosec funtions $\rightarrow$ Relation between arcsin & arccosec funtions </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Relation between arcsin & arccosec funtions </primary>
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- For any $x \geqslant 1$ or $x \leqslant-1 ,\quad$x $ \in \mathbb{R}-(-1,1)$

- $ \operatorname{cosec}^{-1}(x)=\sin ^{-1}\(\frac{1}{x}), |x| \geqslant 1 $

- $\operatorname{cosec}^{-1} x=\theta \quad \theta \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]-\\{0\\}$

- $cosec \left(\operatorname{cosec}^{-1} x\right)=\operatorname{cosec} \theta$

- $ x=\operatorname{cosec} \theta=\frac{1}{\sin \theta}$

- $ \sin \theta=\frac{1}{x}, \frac{1}{x} \leqslant 1, \quad \theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

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<footer style = "font-size: 18px">Inverse trigonometric funtions & their range $\rightarrow$ Inverse trigonometric funtions & their range $\rightarrow$ <span style = "color:blue"> Relation between arcsin & arccosec funtions </span> $\rightarrow$ Relation between arcsin & arccosec funtions $\rightarrow$ Cosine & secant inverse relationship </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Relation between arcsin & arccosec funtions </primary>
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- $ \Rightarrow \sin \theta=\frac{1}{x}, \quad \theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

- $ \sin ^{-1}(\sin \theta)=\sin ^{-1}\left(\frac{1}{x}\right)$

- $ \operatorname{cosec}^{-1} x=\theta=\sin ^{-1}\left(\frac{1}{x}\right)\quad, |x| {\geqslant 1}$

- $ \{cosec}^{-1} x=\sin ^{-1}(\frac{1}{x})$

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<hr>
<footer style = "font-size: 18px">Inverse trigonometric funtions & their range $\rightarrow$ Relation between arcsin & arccosec funtions $\rightarrow$ <span style = "color:blue"> Relation between arcsin & arccosec funtions </span> $\rightarrow$ Cosine & secant inverse relationship $\rightarrow$ Tangent & cotangent inverse relationship </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Cosine & secant inverse relationship </primary>
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- $ \cos ^{-1} \frac{1}{x}=\sec ^{-1} x,|x| \geqslant 1$

- $ \cos ^{-1} \frac{1}{x}=\theta, \quad \theta \in[0, \pi]$      
- $ \cos \left(\cos ^{-1} \frac{1}{x}\right)=\cos \theta$     
- $ \frac{1}{x}=\cos \theta$

- $ \sec \theta=x$    
- Since, $ \theta \in $ Range set of $\sec^{-1},$ $[= \[0, \pi]-\\{\frac{\pi}{2}\\}]$  
- $ \sec ^{-1} x=\theta=\cos ^{-1} \frac{1}{x}$

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<footer style = "font-size: 18px">Relation between arcsin & arccosec funtions $\rightarrow$ Relation between arcsin & arccosec funtions $\rightarrow$ <span style = "color:blue"> Cosine & secant inverse relationship </span> $\rightarrow$ Tangent & cotangent inverse relationship $\rightarrow$ Sine & negative of sine inverse relationship </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Tangent & cotangent inverse relationship </primary>
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- $ \tan ^{-1} \frac{1}{x}=\cot ^{-1} x, \quad x>0$

- Since $x>0, \frac{1}{x}>0$    
- $ \tan ^{-1} \frac{1}{x}=\theta, \theta \in(0, \pi / 2)$   
- $ \tan \theta=\frac{1}{x}$     
- $ \cot \theta=x .$

- Range set of $\cot ^{-1}$ is $(0, \pi)$.     
- $ (0, \pi / 2) \subset(0, \pi), \quad \theta \in(0, \pi)$

- $ \therefore \quad \theta=\cot ^{-1} x .$

- Hence $\tan ^{-1} \frac{1}{x}=\cot ^{-1} x, x>0.$

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<hr>
<footer style = "font-size: 18px">Relation between arcsin & arccosec funtions $\rightarrow$ Cosine & secant inverse relationship $\rightarrow$ <span style = "color:blue"> Tangent & cotangent inverse relationship </span> $\rightarrow$ Sine & negative of sine inverse relationship $\rightarrow$ Tangent inverse as an odd funtion </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Sine & negative of sine inverse relationship </primary>
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- $\sin ^{-1}(-x)$ and $\sin ^{-1}(x)$

- $ x \in[-1,1]$    
- $ \sin ^{-1}(-x)=\theta, \theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$     
- $ \sin \left(\sin ^{-1}(-x)\right)=\sin \theta$    
- $ -x=\sin \theta$     $ [\because \sin (-\theta)=-\sin\theta]$

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

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

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<footer style = "font-size: 18px">Cosine & secant inverse relationship $\rightarrow$ Tangent & cotangent inverse relationship $\rightarrow$ <span style = "color:blue"> Sine & negative of sine inverse relationship </span> $\rightarrow$ Tangent inverse as an odd funtion $\rightarrow$ Cosine  inverse as an odd funtion </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Tangent inverse as an odd funtion </primary>
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- $\tan ^{-1}(-x) \text { and } \tan ^{-1} x $

- $\tan ^{-1}(-x)=\theta, \quad \theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $

- $ \tan \theta=-x $

- $x=-\tan \theta $

- $x=\tan (-\theta) $

- $ -\theta=\tan ^{-1}(x) $

- $ \theta=-\tan ^{-1}(x) $

- $\tan ^{-1}(-x)=-\tan ^{-1}(x)$

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<footer style = "font-size: 18px">Tangent & cotangent inverse relationship $\rightarrow$ Sine & negative of sine inverse relationship $\rightarrow$ <span style = "color:blue"> Tangent inverse as an odd funtion </span> $\rightarrow$ Cosine  inverse as an odd funtion $\rightarrow$ Cosecant inverse as on odd funtion </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Cosine  inverse as an odd funtion </primary>
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- $\cos ^{-1}(-x)$ and $\cos ^{-1} x,|x| \leqslant 1$

- $\cos ^{-1}(-x)=\theta, \quad \theta \in[0, \pi]$

- $\cos \theta=-x $

- $x=-\cos \theta  $

- $x=\cos (\pi-\theta) , \cos (\pi-\theta) =-\cos \theta$

- $ \Rightarrow(\pi-\theta)=\cos ^{-1} x $

- $ \pi-\cos ^{-1}(-x)=\cos ^{-1} x $

- $ \cos ^{-1} x+\cos ^{-1}(-x)=\pi $

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<footer style = "font-size: 18px">Sine & negative of sine inverse relationship $\rightarrow$ Tangent inverse as an odd funtion $\rightarrow$ <span style = "color:blue"> Cosine  inverse as an odd funtion </span> $\rightarrow$ Cosecant inverse as on odd funtion $\rightarrow$ Formula for secant & cotangent inverse functions </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Cosecant inverse as on odd funtion </primary>
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- $\operatorname{cosec}^{-1}(-x)$ and $\operatorname{cosec}^{-1}(x),|x| \geqslant 1$

- $ \operatorname{cosec}^{-1}(-x)=\theta, \quad \theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\\{0\\}$

- $ \operatorname{cosec} \theta=-x$

- $ x=-\operatorname{cosec} \theta$

- $ x=\operatorname{cosec}(-\theta)$

- $ -\theta=\operatorname{cosec}^{-1} x$

- $ \theta=-\operatorname{cosec}^{-1} x$

- $ \operatorname{cosec}^{-1}(-x)=-\operatorname{cosec}^{-1} x$

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<footer style = "font-size: 18px">Tangent inverse as an odd funtion $\rightarrow$ Cosine  inverse as an odd funtion $\rightarrow$ <span style = "color:blue"> Cosecant inverse as on odd funtion </span> $\rightarrow$ Formula for secant & cotangent inverse functions $\rightarrow$ Sine & cosine inverse formula </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Formula for secant & cotangent inverse functions </primary>
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- $\rightarrow \sec ^{-1}(-x)=\pi-\sec ^{-1}(x), \quad|x| \geqslant 1$

- $\rightarrow \cot ^{-1}(-x)=\pi-\cot ^{-1}(x), \quad x \in \mathbb{R}$

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<footer style = "font-size: 18px">Cosine  inverse as an odd funtion $\rightarrow$ Cosecant inverse as on odd funtion $\rightarrow$ <span style = "color:blue"> Formula for secant & cotangent inverse functions </span> $\rightarrow$ Sine & cosine inverse formula $\rightarrow$ Some other formulas </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Sine & cosine inverse formula </primary>
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- $\sin ^{-1} x$ and $\cos ^{-1} x,|x| \leqslant 1$

- $ \Rightarrow \sin ^{-1} x=\theta, \theta \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$    
- $ \sin \left(\sin ^{-1} x\right)=\sin \theta$    
- $ x=\sin \theta$

- $ \Rightarrow x=\cos \left(\frac{\pi}{2}-\theta\right) [\because\sin \theta=\cos \left(\frac{\pi}{2}-\theta\right)]$

- $  \frac{\pi}{2}-\theta \in[0, \pi]$     
- $ {\Rightarrow} \pi / 2-\theta=\cos ^{-1} x$    
- $ \pi / 2-\sin ^{-1} x=\cos ^{-1} x$    
- $ \sin ^{-1} x+\cos ^{-1} x=\pi / 2$

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<footer style = "font-size: 18px">Cosecant inverse as on odd funtion $\rightarrow$ Formula for secant & cotangent inverse functions $\rightarrow$ <span style = "color:blue"> Sine & cosine inverse formula </span> $\rightarrow$ Some other formulas $\rightarrow$ Tangent inverse formula </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Tangent inverse formula </primary>
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<main>

- We need to find out a formula something like this,

- $ \tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}(?) $

- Let, $ \tan ^{-1} x=\theta, \tan ^{-1} y=\phi$

- where, $ \theta, \phi \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$  $ [x=\tan \theta,y=\tan\phi]$

- $\Rightarrow (\theta+\phi)=\tan ^{-1}(?)$

- $ \tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}=\frac{x+y}{1-x y}$

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</main>
<hr>
<footer style = "font-size: 18px">Sine & cosine inverse formula $\rightarrow$ Some other formulas $\rightarrow$ <span style = "color:blue"> Tangent inverse formula </span> $\rightarrow$ Tangent inverse formula $\rightarrow$ Important conditions for  $\tan ^{-1} x+\tan ^{-1} y$ </footer>

### <Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success>
### <primary style="font-size:24px"> Tangent inverse formula </primary>
<hr>
<main>

- Taking $\tan^{-1}$ both side we get,

- $ (\theta+\phi)=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$

- If $ (\theta+\phi) \in(\frac{-\pi}{2}, \frac{\pi}{2})$ $\in xy<1$

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

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</main>
<hr>
<footer style = "font-size: 18px">Some other formulas $\rightarrow$ Tangent inverse formula $\rightarrow$ <span style = "color:blue"> Tangent inverse formula </span> $\rightarrow$ Important conditions for  $\tan ^{-1} x+\tan ^{-1} y$ $\rightarrow$ Thankyou </footer>

<h3 id="success-stylefont-size25px-inverse-trigonometric-functions-l-2-success-3"><Success style="font-size:25px"> Inverse Trigonometric Functions L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-important-conditions-for--tan--1-xtan--1-y-primary"><primary style="font-size:24px"> Important conditions for  $\tan ^{-1} x+\tan ^{-1} y$ </primary></h3>
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<div style='float: left; width:99%; font-size:18px'>
<table>
<thead>
<tr>
<th>CONDITION</th>
<th>$$\tan ^{-1} x+\tan ^{-1} y$$</th>
</tr>
</thead>
<tbody>
<tr>
<td>$x y<1$</td>
<td>$$\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$$</td>
</tr>
<tr>
<td>$x>0, y>0, x y>1$</td>
<td>$$\pi+\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$$</td>
</tr>
<tr>
<td>$x<0, y<0, x y>1$</td>
<td>$$-\pi+\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$$</td>
</tr>
</tbody>
</table>
</div>
</main>
<hr>
<footer style = "font-size: 18px">Tangent inverse formula $\rightarrow$ Tangent inverse formula $\rightarrow$ <span style = "color:blue"> Important conditions for  $\tan ^{-1} x+\tan ^{-1} y$ </span> $\rightarrow$ Thankyou  $\rightarrow$ </footer>