1. Values of trigonometrical ratios of some particular angles

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• $\quad \tan \mathrm{A}+\tan \mathrm{B}+\tan \mathrm{C}=\tan \mathrm{A} \tan B \tan \mathrm{C}$

• $\quad \cot \mathrm{A} \cot \mathrm{B}+\cot \mathrm{B} \cot \mathrm{C}+\cot C \cot \mathrm{A}=1$

• $\quad \tan \frac{A}{2} \tan \frac{B}{2}+\tan \frac{B}{2} \tan \frac{C}{2}+\tan \frac{C}{2} \tan \frac{A}{2}=1$

• $\quad \cot \frac{\mathrm{A}}{2}+\cot \frac{\mathrm{B}}{2}+\cot \frac{\mathrm{C}}{2}=\cot \frac{\mathrm{A}}{2} \cot \frac{\mathrm{B}}{2} \cot \frac{\mathrm{C}}{2}$

• $\quad \sin 2 \mathrm{~A}+\sin 2 \mathrm{~B}+\sin 2 \mathrm{C}=4 \sin \mathrm{A} \sin B \sin \mathrm{C}$

• $\quad \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}=-1-4 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}$

• $\quad \sin \mathrm{A}+\sin \mathrm{B}+\sin \mathrm{C}=4 \cos \frac{\mathrm{A}}{2} \cos \frac{\mathrm{B}}{2} \cos \frac{\mathrm{C}}{2}$

• $\quad \cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}=1+4 \sin \frac{\mathrm{A}}{2} \sin \frac{\mathrm{B}}{2} \sin \frac{\mathrm{C}}{2}$

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

• $\quad \sin \left(\mathrm{A} _1+\mathrm{A} _2 \ldots \ldots \ldots \ldots \ldots . .+\mathrm{A} _{\mathrm{n}}\right) \quad=\cos \mathrm{A} _1 \cos \mathrm{A} _2 \ldots \ldots \ldots \ldots . \cos \mathrm{A} _{\mathrm{n}}\left(\mathrm{S} _1-\mathrm{S} _3+\mathrm{S}_5-\ldots \ldots \ldots \ldots \ldots.\right)$

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

• $\quad \tan \left(\mathrm{A} _1+\mathrm{A} _2 \ldots \ldots \ldots \ldots \ldots \ldots+\mathrm{A} _{\mathrm{n}}\right) \quad=\frac{\mathrm{S} _1-\mathrm{S}_3+\mathrm{S}_5-\ldots \ldots . .}{1-\mathrm{S}_2+\mathrm{S}_4-\mathrm{S}_6+\ldots \ldots}$

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

$\mathrm{S} _{2}=\sum \tan \mathrm{A} _{1} \tan \mathrm{A} _{2}=$ 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.

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

$\cos \mathrm{nA}=\cos ^{\mathrm{n}} \mathrm{A}\left(1-{ }^{\mathrm{n}} \mathrm{C} _{2} \tan ^{2} \mathrm{~A}+{ }^{\mathrm{n}} \mathrm{C} _{4} \tan ^{4} \mathrm{~A}-……..\right)$

$\tan \mathrm{nA}=\frac{{ }^{\mathrm{n}} \mathrm{C} _{1} \tan \mathrm{A}-{ }^{\mathrm{n}} \mathrm{C} _{3} \tan ^{3} \mathrm{~A}+{ }^{\mathrm{n}} \mathrm{C} _{5} \tan ^{5} \mathrm{~A}-\ldots \ldots \ldots . . . .}{1-{ }^{\mathrm{n}} \mathrm{C} _{2} \tan ^{2} \mathrm{~A}+{ }^{\mathrm{n}} \mathrm{C} _{4} \tan ^{4} \mathrm{~A}-\ldots \ldots \ldots \ldots \ldots \ldots . . . . . . .}$

Solved examples

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

(a). $2$

(b). $-\frac{1}{2}$

(c). $\frac{1}{2}$

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

(a). $\sqrt{13}$

(b). $\sqrt{14}$

(c). $\sqrt{17}$

(d). $\sqrt{15}$

5. If $\alpha+\beta=\frac{\pi}{2}$ 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. $\sum _{\mathrm{r}=1}^{7} \tan ^{2} \frac{\mathrm{r} \pi}{16}=$

(a). 34

(b). 35

(c). 37

(d). None of these

Exercise

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

(a). $1$

(b). $-1$

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

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

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). $\mathrm{k} \leq \frac{1}{3 \sqrt{3}}$

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

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

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

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

(a). $\frac{1}{x y z}$

(b). $\frac{4}{x y z}$

(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). $\frac{1}{3}$

(c). $\frac{-1}{3}$

(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 $\operatorname{sinn} \theta=\sum _{\mathrm{r}=0}^{\mathrm{n}} \mathrm{b} _{\mathrm{r}} \sin ^{\mathrm{r}} \theta, \forall \theta$, 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 $\mathrm{e}^{-\pi / 2}<\theta<\frac{\pi}{2}$, 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). $\log _{e}(\cos \theta)$

(c). both are equal

(d). None of these

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

(a). $\frac{n}{2}$

(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 $\sum \frac{\sqrt{\sin A}}{\sqrt{\sin B}+\sqrt{\sin C}-\sqrt{\sin A}}$ is

(a). 3

(b). 0

(c). 4

(d). None of these

12. If $\cos \frac{\pi}{7}, \cos \frac{3 \pi}{7}, \cos \frac{5 \pi}{7}$, 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 $\sec \frac{\pi}{7}+\sec \frac{3 \pi}{7}+\sec \frac{5 \pi}{7}$ is

(a). 2

(b). 4

(c). 8

(d). None of these

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

(a). $\frac{1}{4}$

(b). $\frac{1}{8}$

(c). $\frac{\sqrt{7}}{4}$

(d). $\frac{\sqrt{7}}{8}$

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

(a). $\frac{1}{4}$

(b). $\frac{1}{8}$

(c). $\frac{\sqrt{7}}{4}$

(d). $\frac{\sqrt{7}}{8}$

(iv). The equation whose roots $\operatorname{arc}^{2} \tan ^{2} \frac{\pi}{7}, \tan ^{2} \frac{3 \pi}{7}, \& \tan ^{2} \frac{5 \pi}{7}$, 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 $\sum _{\mathrm{r}=1}^{3} \tan ^{2}\left(\frac{2 \mathrm{r}-1}{7}\right) \sum _{\mathrm{r}=1}^{3} \cot ^{2}\left(\frac{2 \mathrm{r}-1}{7}\right)$ is

(a). 15

(b). 105

(c). 21

(d). 147

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

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

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

(c). $2[x]$

(d). $[\mathrm{x}]^{2}$

14. If $\tan \alpha, \tan \beta, \tan \gamma$ are the roots of $\mathrm{x}^{3}-\mathrm{px}^{2}-\mathrm{r}=0$, 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). $2$