1. Values of trigonometrical ratios of some particular angles

(i) $\sin 7 \dfrac{1}{2}^{\circ}=\dfrac{\sqrt{4-\sqrt{2}-\sqrt{6}}}{2 \sqrt{2}}$

$\cos 7 \dfrac{1}{2}^{\circ}=\dfrac{\sqrt{4+\sqrt{2}+\sqrt{6}}}{2 \sqrt{2}}$

$\tan 7 \dfrac{1}{2}^{\circ}=(\sqrt{3}-\sqrt{2})(\sqrt{2}-1)$

$\cot 7 \dfrac{1}{2}^{\circ}=(\sqrt{3}+\sqrt{2})(\sqrt{2}+1)$

(ii) $\sin 15^{\circ}=\cos 75^{\circ}=\dfrac{\sqrt{3}-1}{2 \sqrt{2}}$

$\cos 15^{\circ}=\sin 75^{\circ}=\dfrac{\sqrt{3}+1}{2 \sqrt{2}}$

$\tan 15^{\circ}=\cot 75^{\circ}=2-\sqrt{3}$

$\cot 15^{\circ}=\tan 75^{\circ}=2+\sqrt{3}$

(iii) $\sin 22 \dfrac{1}{2}^{\circ}=\dfrac{1}{2} \sqrt{2-\sqrt{2}}$

$\cos 22 \dfrac{1}{2}^{\circ}=\dfrac{1}{2} \sqrt{2+\sqrt{2}}$

$\tan 22 \dfrac{1}{2}^{\circ}=\sqrt{2}-1$

$\cot 22 \dfrac{1}{2}^{\circ}=\sqrt{2}+1$

(iv) $\sin 18^{\circ}=\cos 72^{\circ}=\dfrac{\sqrt{5}-1}{4}$

$\cos 18^{\circ}=\sin 72^{\circ}=\dfrac{\sqrt{10+2 \sqrt{5}}}{4}$

$\sin 36^{\circ}=\cos 54^{\circ}=\dfrac{\sqrt{10-2 \sqrt{5}}}{4}$

$\cos 36^{\circ}=\sin 54^{\circ}=\dfrac{\sqrt{5}+1}{4}$

(v) $\cos 9^{\circ}=\frac{1}{2}\left(\sqrt{1+\sin 18^{\circ}}+\sqrt{1-\sin 18^{\circ}}\right)$

(vi) $\cos 27^{\circ}=\dfrac{1}{2}\left(\sqrt{1+\cos 36^{\circ}}+\sqrt{1-\cos 36^{\circ}}\right)$

2. Conditional identities

If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are angles of a triangle (i.e. $\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$ ) then

• $\tan \mathrm{A}+\tan \mathrm{B}+\tan \mathrm{C}=\tan \mathrm{A} \tan B \tan \mathrm{C}$
• $\cot \mathrm{A} \cot \mathrm{B}+\cot \mathrm{B} \cot \mathrm{C}+\cot \mathrm{C} \cot \mathrm{A}=1$
• $\tan \dfrac{A}{2} \tan \dfrac{B}{2}+\tan \dfrac{B}{2} \tan \dfrac{C}{2}+\tan \dfrac{C}{2} \tan \dfrac{A}{2}=1$
• $\cot \dfrac{\mathrm{A}}{2}+\cot \dfrac{\mathrm{B}}{2}+\cot \dfrac{\mathrm{C}}{2}=\cot \dfrac{\mathrm{A}}{2} \cot \dfrac{\mathrm{B}}{2} \cot \dfrac{\mathrm{C}}{2}$
• $\sin 2 \mathrm{~A}+\sin 2 \mathrm{~B}+\sin 2 \mathrm{C}=4 \sin \mathrm{A} \sin B \sin \mathrm{C}$
• $\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}=-1-4 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}$
• $\sin \mathrm{A}+\sin \mathrm{B}+\sin \mathrm{C}=4 \cos \dfrac{\mathrm{A}}{2} \cos \dfrac{\mathrm{B}}{2} \cos \dfrac{\mathrm{C}}{2}$
• $\cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}=1+4 \sin \dfrac{\mathrm{A}}{2} \sin \dfrac{\mathrm{B}}{2} \sin \dfrac{\mathrm{C}}{2}$

3. Trigonometric ratios of sum of more than three angles.

• $\sin \left(\mathrm{A} _{1}+\mathrm{A} _{2}\right.$ ………………..$\left.+\mathrm{A} _{\mathrm{n}}\right) \quad=\cos \mathrm{A} _{1} \cos \mathrm{A}$…………………$\cos \mathrm{A} _{\mathrm{n}}\left(\mathrm{S} _{1}-\mathrm{S} _{3}+\mathrm{S} _{5}-\ldots \ldots \ldots \ldots \ldots .\right.)$

• $\cos \left(\mathrm{A} _{1}+\mathrm{A} _{2}\right.$…………………..$\left.+\mathrm{A} _{\mathrm{n}}\right) \quad=\cos \mathrm{A} _{1} \cos \mathrm{A}$………………$\cos \mathrm{A} _{\mathrm{n}}\left(1-\mathrm{S} _{2}+\mathrm{S} _{4}-\mathrm{S} _{6}+……………..\right.)$

• $\tan \left(\mathrm{A} _{1}+\mathrm{A}\right.$………………..$\left.+A _{n}\right)=\dfrac{S _{1}-S _{3}+S _{5}-\ldots \ldots}{1-S _{2}+S _{4}-S _{6}+\ldots . .}$

where $S _{1}=\sum \tan A _{1} \quad=$ sum of tangents of angles

$\hspace {1 cm}\mathrm{S} _{2}=\sum \tan \mathrm{A} _{1} \tan \mathrm{A} _{2}=$ sum of tangents taken two at a time etc.

In particular, if $\mathrm{A} _{1}=\mathrm{A} _{2}=$………………..$A _{n}=A$, then

$\mathrm{S} _{1}=\mathrm{n} \tan \mathrm{A} ; \mathrm{S} _{2}={ }^{n} \mathrm{C} _{2} \tan ^{2} \mathrm{~A} ; \mathrm{S} _{3}={ }^{n} \mathrm{C} _{3} \tan ^{3} \mathrm{~A}$ etc.

$\sin \mathrm{nA}=\cos ^{\mathrm{n}} \mathrm{A}\left({ }^{\mathrm{n}} \mathrm{C} _{1} \tan \mathrm{A}-{ }^{\mathrm{n}} \mathrm{C} _{3} \tan ^{3} \mathrm{~A}+{ }^{+} \mathrm{C} _{5} \tan ^{5} \mathrm{~A}-\right.$………………..)

$\cos \mathrm{n} A=\cos ^{\mathrm{n}} \mathrm{A}\left(1-{ }^{\mathrm{n}} \mathrm{C} _2 \tan ^2 \mathrm{~A}+{ }^{\mathrm{n}} \mathrm{C} _4 \tan ^4 \mathrm{~A}-\ldots \ldots \ldots \ldots \ldots \ldots\right)$

$\tan \mathrm{n} \mathrm{A}=\dfrac{{ }^{\mathrm{n}} \mathrm{C} _1 \tan \mathrm{A}-{ }^{\mathrm{n}} \mathrm{C} _3 \tan ^3 \mathrm{~A}+{ }^n \mathrm{C} _5 \tan ^5 \mathrm{~A}-\ldots \ldots \ldots .}{1-{ }^{\mathrm{n}} \mathrm{C} _2 \tan ^2 \mathrm{~A}+{ }^{\mathrm{n}} \mathrm{C} _4 \tan ^4 \mathrm{~A}-\ldots \ldots \ldots \ldots \ldots \ldots .}$

Solved Examples

1. If $f(\mathrm{x})=\dfrac{\cot \mathrm{x}}{1+\cot \mathrm{x}}$ and $\alpha+\beta=\dfrac{5 \pi}{4}$, then the value of $f(\alpha) \cdot f(\beta)$ is

(a) 2

(b) $-\dfrac{1}{2}$

(c) $\dfrac{1}{2}$

(d) None of these

2. The value of $\tan 81^{\circ}-\tan 63^{\circ}-\tan 27^{\circ}+\tan 9^{\circ}$ equals

(a) 1

(b) 2

(c) 3

(d) 4

3. The number of integral values of $k$ for which the equation $7 \cos x+5 \sin x=2 k+1$ has a unique solution is

(a) 4

(b) 8

(c) 10

(d) 12

4. If $\dfrac{\sin x}{\sin y}=\dfrac{1}{2}$ and $\dfrac{\cos x}{\cos y}=\dfrac{3}{2}$ where $x, y \in\left(0, \dfrac{\pi}{2}\right)$ then $\tan (x+y)=$

(a) $\sqrt{13}$

(b) $\sqrt{14}$

(c) $\sqrt{17}$

(d) $\sqrt{15}$

5. If $\alpha+\beta=\dfrac{\pi}{2}$ and $\beta+\gamma=\alpha$, then $\tan \alpha$ is equal to

(a) $2(\tan \beta+\tan \gamma)$

(b) $\tan \beta+\tan \gamma$

(c) $\tan \beta+2 \tan \gamma$

(d) $2 \tan \beta+\tan \gamma$

6. $\sum\limits _{\mathrm{r}=1}^{7} \tan ^{2} \dfrac{\mathrm{r} \pi}{16}=$

(a) 34

(b) 35

(c) 37

(d) None of these

Exercise

1. If $\mathrm{p} _{\mathrm{n}+1}=\sqrt{\dfrac{1}{2}\left(1+\mathrm{p} _{\mathrm{n}}\right)}$, then $\cos \left(\dfrac{\sqrt{1-\mathrm{p} _{0}{ }^{2}}}{\mathrm{p} _{1} \mathrm{p} _{2} \mathrm{p} _{3} \ldots \ldots \infty}\right)$ is equal to

(a) 1

(b) -1

(c) $\mathrm{p} _{0}$

(d) $\dfrac{1}{\mathrm{p} _{0}}$

2. If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are acute positive angles such that $\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$ and $\cot \mathrm{A} \cot \mathrm{B} \cot \mathrm{C}=\mathrm{k}$, then

(a) $\mathrm{k} \leq \dfrac{1}{3 \sqrt{3}}$

(b) $\mathrm{k} \geq \dfrac{1}{3 \sqrt{3}}$

(c) $\mathrm{k}<\dfrac{1}{9}$

(d) $\mathrm{k}>\dfrac{1}{3}$

3. If $\sum x y=1$, then $\sum \dfrac{x+y}{1-x y}=$

(a) $\dfrac{1}{x y z}$

(b) $\dfrac{4}{x y z}$

(c) $x y z$

(d) None of these

4. The value of $\cot 16^{\circ} \cot 44^{\circ}+\cot 44^{\circ} \cot 76^{\circ}-\cot 76^{\circ} \cot 16^{\circ}$ is

(a) 3

(b) $\dfrac{1}{3}$

(c) $\dfrac{-1}{3}$

(d) -3

5. The number of solutions of $\tan (5 \pi \cos \theta)=\cot (5 \pi \sin \theta)$ for $\theta$ in $(0,2 \pi)$ is

(a) 28

(b) 14

(c) 4

(d) 2

6. If $\cos x=\tan y, \cos y=\tan z$ and $\cos z=\tan x$, then a value of $\sin x$ is equal to

(a) $2 \cos 18^{\circ}$

(c) $\sin 18^{\circ}$

(b) $\cos 18^{\circ}$

(d) $2 \sin 18^{\circ}$

7. Let $n$ be an odd integer. If $\operatorname{sinn} \theta=\sum\limits _{\mathrm{r}=0}^{\mathrm{n}} \mathrm{b} _{\mathrm{r}} \sin ^{\mathrm{r}} \theta, \forall \theta$, then

(a) $\mathrm{b} _{0}=1, \mathrm{~b} _{1}=3$

(c) $\mathrm{b} _{0}=-1 \mathrm{~b} _{1}=\mathrm{n}$

(b) $\mathrm{b} _{0}=0, \mathrm{~b} _{1}=\mathrm{n}$

(d) $\mathrm{b} _{0}=0, \mathrm{~b} _{1}=\mathrm{n}^{2}-3 \mathrm{n}+3$

8. If $\mathrm{e}^{-\pi / 2}<\theta<\dfrac{\pi}{2}$, which is larger, $\cos \left(\log _{\mathrm{e}} \theta\right)$ or $\log _{\mathrm{e}}(\cos \theta)$

(a) $\cos \left(\log _{\mathrm{e}} \theta\right)$

(b) $\log _{\mathrm{e}}(\cos \theta)$

(c) both are equal

(d) None of these

9. $\quad \sum\limits _{r=1}^{n-1}(n-r) \cos \dfrac{2 r \pi}{n}$ for $n \geq 3$ is

(a) $\dfrac{n}{2}$

(b) $\mathrm{n}$

(c) $(\mathrm{n}-3)$

(d) None of these

10.* Match the following :-

11. In any $\triangle A B C$, the minimum value of $\sum \dfrac{\sqrt{\sin A}}{\sqrt{\sin B}+\sqrt{\sin C}-\sqrt{\sin A}}$ is

(a) 3

(b) 0

(c) 4

(d) None of these

12. If $\cos \dfrac{\pi}{7}, \cos \dfrac{3 \pi}{7}, \cos \dfrac{5 \pi}{7}$, are the roots of the equation $8 x^{3}-4 x^{2}-4 x+1=0$.

On the basis of above information, answer the following questions :-

(i) The value of $\sec \dfrac{\pi}{7}+\sec \dfrac{3 \pi}{7}+\sec \dfrac{5 \pi}{7}$ is

these

(a) 2

(b) 4

(c) 8

(d) None of

(ii) The value of $\sin \dfrac{\pi}{14} \sin \dfrac{3 \pi}{14} \sin \dfrac{5 \pi}{14}$ is

(a) $\dfrac{1}{4}$

(b) $\dfrac{1}{8}$

(c) $\dfrac{\sqrt{7}}{4}$

(d) $\dfrac{\sqrt{7}}{8}$

(iii) The value of $\cos \dfrac{\pi}{14} \cos \dfrac{3 \pi}{14} \cos \dfrac{5 \pi}{14}$ is

(a) $\dfrac{1}{4}$

(b) $\dfrac{1}{8}$

(c) $\dfrac{\sqrt{7}}{4}$

(d) $\dfrac{\sqrt{7}}{8}$

(iv) The equation whose roots $\operatorname{arc}^{2} \tan ^{2} \dfrac{\pi}{7}, \tan ^{2} \dfrac{3 \pi}{7}, \& \tan ^{2} \dfrac{5 \pi}{7}$, is

(a) $x^{3}-35 x^{2}+7 x-21=0$

(b) $x^{3}-35 x^{2}+21 x-7=0$

(c) $x^{3}-21 x^{2}+35 x-7=0$

(d) $x^{3}-21 x^{2}+7 x-35=0$

(v) the value of $\sum\limits _{\mathrm{r}=1}^{3} \tan ^{2}\left(\dfrac{2 \mathrm{r}-1}{7}\right) \sum\limits _{\mathrm{r}=1}^{3} \cot ^{2}\left(\dfrac{2 \mathrm{r}-1}{7}\right)$ is

(a) 15

(b) 105

(c) 21

(d) 147

13. If $a=\sin \dfrac{\pi}{18} \sin \dfrac{5 \pi}{18} \sin \dfrac{7 \pi}{18}$, and $x$ is the solution of the equation $y=2[x]+2$ and $y=3[x-2]$, then $\mathrm{a}=$

(a) $[\mathrm{x}]$

(b) $\dfrac{1}{[\mathrm{x}]}$

(c) $2[x]$

(d) $[x]^{2}$

14. If $\tan \alpha, \tan \beta, \tan \gamma$ are the roots of $x^{3}-p^{2}-r=0$, then the value of $\left(1+\tan ^{2} \alpha\right)\left(1+\tan ^{2} \beta\right)\left(1+\tan ^{2} \gamma\right)$ is equal to

(a) $(\mathrm{p}-\mathrm{r})^{2}$

(b) $1+(\mathrm{p}-\mathrm{r})^{2}$

(c) $1-(\mathrm{p}-\mathrm{r})^{2}$

(d) None of these

15. If $\tan \alpha$ is an integral solution of $4 x^{2}-16 x+15<0$ and $\cos \beta$ is the slope of the bisector of the angle in the first quadrant between the $x \& y$ axes, then the value of $\sin (\alpha+\beta): \sin (\alpha-\beta)$ is equal to

(a) -1

(b) 0

(c) 1

(d) 2