1. Values of trigonometrical ratios of some particular angles

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(vi) $\cos {27}^{\circ }={\displaystyle \frac{1}{2}}(\sqrt{1+\cos {36}^{\circ }}+\sqrt{1-\cos {36}^{\circ }})$

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

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

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

• $\sin ({\mathrm{A}}_1+{\mathrm{A}}_2$ ………………..$+{\mathrm{A}}_{\mathrm{n}})=\cos {\mathrm{A}}_1\cos \mathrm{A}$…………………$\cos {\mathrm{A}}_{\mathrm{n}}({\mathrm{S}}_1-{\mathrm{S}}_3+{\mathrm{S}}_5-\dots \dots \dots \dots \dots .)$

• $\cos ({\mathrm{A}}_1+{\mathrm{A}}_2$…………………..$+{\mathrm{A}}_{\mathrm{n}})=\cos {\mathrm{A}}_1\cos \mathrm{A}$………………$\cos {\mathrm{A}}_{\mathrm{n}}(1-{\mathrm{S}}_2+{\mathrm{S}}_4-{\mathrm{S}}_6+\dots \dots \dots \dots \dots ..)$

• $\tan ({\mathrm{A}}_1+\mathrm{A}$………………..$+A_n)={\displaystyle \frac{S_1-S_3+S_5-\dots \dots }{1-S_2+S_4-S_6+\dots ..}}$

where $S_1=\sum \tan A_1=$ 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.

$\sin \mathrm{nA}={\cos }^{\mathrm{n}}\mathrm{A}({}^{\mathrm{n}}{\mathrm{C}}_1\tan \mathrm{A}-{}^{\mathrm{n}}{\mathrm{C}}_3{\tan }^3\mathrm{A}+{}^{+}{\mathrm{C}}_5{\tan }^5\mathrm{A}-$………………..)

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

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

Solved Examples

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

(a) 2

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

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

(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 $7\cos x+5\sin x=2k+1$ has a unique solution is

(a) 4

(b) 8

(c) 10

(d) 12

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

(a) $\sqrt{13}$

(b) $\sqrt{14}$

(c) $\sqrt{17}$

(d) $\sqrt{15}$

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

(a) 34

(b) 35

(c) 37

(d) None of these

Exercise

1. If ${\mathrm{p}}_{\mathrm{n}+1}=\sqrt{{\displaystyle \frac{1}{2}}(1+{\mathrm{p}}_{\mathrm{n}})}$, then $\cos \left({\displaystyle \frac{\sqrt{1-{\mathrm{p}}_0{}^2}}{{\mathrm{p}}_1{\mathrm{p}}_2{\mathrm{p}}_3\dots \dots \mathrm{\infty }}}\right)$ is equal to

(a) 1

(b) -1

(c) ${\mathrm{p}}_0$

(d) ${\displaystyle \frac{1}{{\mathrm{p}}_0}}$

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) $\mathrm{k}\le {\displaystyle \frac{1}{3\sqrt{3}}}$

(b) $\mathrm{k}\ge {\displaystyle \frac{1}{3\sqrt{3}}}$

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

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

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

(a) ${\displaystyle \frac{1}{xyz}}$

(b) ${\displaystyle \frac{4}{xyz}}$

(c) $xyz$

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

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

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

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

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

(c) both are equal

(d) None of these

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

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

(b) $\mathrm{n}$

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

(d) None of these

10.* Match the following :-

11. In any $\mathrm{△}ABC$, the minimum value of $\sum {\displaystyle \frac{\sqrt{\sin A}}{\sqrt{\sin B}+\sqrt{\sin C}-\sqrt{\sin A}}}$ is

(a) 3

(b) 0

(c) 4

(d) None of these

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

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

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

these

(a) 2

(b) 4

(c) 8

(d) None of

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(a) 15

(b) 105

(c) 21

(d) 147

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

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

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

(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 $(1+{\tan }^2\alpha )(1+{\tan }^2\beta )(1+{\tan }^2\gamma )$ 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 $4x^2-16x+15<0$ and $\cos \beta$ is the slope of the bisector of the angle in the first quadrant between the $x\mathrm{\&}y$ axes, then the value of $\sin (\alpha +\beta ):\sin (\alpha -\beta )$ is equal to

(a) -1

(b) 0

(c) 1

(d) 2