Chapter 11

$$ \text { Find the value of } \sin 10^{\circ}+\sin 20^{\circ}+\sin 30^{\circ}+\cdots \cdots+\sin 360^{\circ} $$

$$ \text { If } x=y \cos \frac{2 \pi}{3}=z \cos \frac{4 \pi}{3}, \text { then show that } x y+y z+z x=0 $$

$$ \cos \left(45^{\circ}-A\right) \cos \left(45^{\circ}-B\right)-\sin \left(45^{\circ}-A\right) \sin \left(45^{\circ}-B\right)=\sin (A+B) $$

$$ \sin \left(45^{\circ}+A\right) \cos \left(45^{\circ}-B\right)+\cos \left(45^{\circ}+A\right) \sin \left(45^{\circ}-B\right)=\cos (A-B) $$

$$ \frac{\sin (A-B)}{\cos A \cos B}+\frac{\sin (B-C)}{\cos B \cos C}+\frac{\sin (C-A)}{\cos C \cos A}=0 $$

$$ \cos A \cos (B-A)-\sin A \sin (B-A)=\cos B $$

$$ \cos (A+B) \cos C-\cos (B+C) \cos A=\sin B \sin (C-A) $$

$$ \sin (n+1) A \sin (n-1) A+\cos (n+1) A \cos (n-1) A=\cos 2 A $$

$$ \sin (n+1) A \sin (n+2) A+\cos (n+1) A \cos (n+2) A=\cos A $$

$$ \sin ^{2}\left(\frac{\pi}{8}+\frac{A}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{A}{2}\right)=\frac{1}{\sqrt{2}} \sin A $$

$$ \cos 2 \theta \cos 2 \phi+\sin ^{2}(\theta-\phi)-\sin ^{2}(\theta+\phi)=\cos (2 \theta+2 \phi) $$

$$ \cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ}=\frac{\sqrt{5}+1}{8} $$

$$ \sin ^{2} 24^{\circ}-\sin ^{2} 6^{\circ}=\frac{\sqrt{5}-1}{8} $$

$$ \cot 16^{\circ} \cot 44^{\circ}+\cot 44^{\circ} \cot 76^{\circ}-\cot 76^{\circ} \cot 16^{\circ}=3 $$

$$ \text { If } \frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}, \text { then find the value of } \frac{\tan x}{\tan y} \text { in terms of } a \text { and } b \text { . } $$

$$ \text { If } \tan A=\frac{1}{2}, \tan B=\frac{1}{3}, \text { show that } A+B=45^{\circ} $$

$$ \text { If } \tan A=\frac{m}{m+1}, \tan B=\frac{1}{2 m+1}, \text { show that } A+B=45^{\circ} \text { . } $$

$$ \frac{\sin 7 \theta-\sin 5 \theta}{\cos 7 \theta+\cos 5 \theta}=\tan \theta $$

$$ \frac{\cos 6 \theta-\cos 4 \theta}{\sin 6 \theta+\sin 4 \theta}=-\tan \theta $$

$$ \frac{\sin A+\sin 3 A}{\cos A+\cos 3 A}=\tan 2 A $$

$$ \frac{\sin 7 A-\sin A}{\sin 8 A-\sin 2 A}=\cos 4 A \sec 5 A \text { . } $$

$$ \frac{\cos 2 B+\cos 2 A}{\cos 2 B-\cos 2 A}=\cot (A+B) \cot (A-B) $$

$$ \frac{\sin 2 A+\sin 2 B}{\sin 2 A-\sin 2 B}=\frac{\tan (A+B)}{\tan (A-B)} $$

$$ \frac{\sin A+\sin 2 A}{\cos A-\cos 2 A}=\cot \frac{A}{2} $$

$$ \frac{\sin 5 A-\sin 3 A}{\cos 3 A+\cos 5 A}=\tan A $$

$$ \frac{\cos 2 B-\cos 2 A}{\sin 2 B+\sin 2 A}=\tan (A-B) $$

$$ \cos (A+B)+\sin (A-B)=2 \sin \left(45^{\circ}+A\right) \cos \left(45^{\circ}+B\right) . $$

$$ \frac{\cos 3 A-\cos A}{\sin 3 A-\sin A}+\frac{\cos 2 A-\cos 4 A}{\sin 4 A-\sin 2 A}=\frac{\sin A}{\cos 2 A \cos 3 A} $$

$$ \frac{\sin (4 A-2 B)+\sin (4 B-2 A)}{\cos (4 A-2 B)+\cos (4 B-2 A)}=\tan (A+B) $$

$$ \frac{\cos 3 \theta+2 \cos 5 \theta+\cos 7 \theta}{\cos \theta+2 \cos 3 \theta+\cos 5 \theta}=\cos 2 \theta-\sin 2 \theta \tan 3 \theta . $$

$$ \frac{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}=\tan 4 A $$

$$ \frac{\sin (\theta+\phi)-2 \sin \theta+\sin (\theta-\phi)}{\cos (\theta+\phi)-2 \cos \theta+\cos (\theta-\phi)}=\tan \theta $$

$$ \frac{\sin A+2 \sin 3 A+\sin 5 A}{\sin 3 A+2 \sin 5 A+\sin 7 A}=\frac{\sin 3 A}{\sin 5 A} . $$

$$ \frac{\sin (A-C)+2 \sin A+\sin (A+C)}{\sin (B-C)+2 \sin B+\sin (B+C)}=\frac{\sin A}{\sin B} $$

$$ \frac{\sin A-\sin 5 A+\sin 9 A-\sin 13 A}{\cos A-\cos 5 A-\cos 9 A+\cos 13 A}=\cot 4 A $$

$$ \frac{\sin A+\sin B}{\sin A-\sin B}=\tan \frac{A+B}{2} \cot \frac{A-B}{2} $$

$$ \frac{\cos A+\cos B}{\cos B-\cos A}=\cot \frac{A+B}{2} \cot \frac{A-B}{2} $$

$$ \frac{\sin A+\sin B}{\cos A+\cos B}=\tan \frac{A+B}{2} $$

$$ \frac{\sin A-\sin B}{\cos B-\cos A}=\cot \frac{A+B}{2} $$

$$ \frac{\cos (A+B+C)+\cos (-A+B+C)+\cos (A-B+C)+\cos (A+B-C)}{\sin (A+B+C)+\sin (-A+B+C)-\sin (A-B+C)+\sin (A+B-C)}=\cot B $$

$$ \cos 3 A+\cos 5 A+\cos 7 A+\cos 15 A=4 \cos 4 A \cos 5 A \cos 6 A $$

$$ \sin a+\sin 2 a+\sin 4 a+\sin 5 a=4 \cos \frac{a}{2} \cos \frac{3 a}{2} \sin 3 a $$

$$ \cos (-A+B+C)+\cos (A-B+C)+\cos (A+B-C)+\cos (A+B+C)=4 \cos A \cos B \cos C $$

$$ \sin (A+B+C+D)+\sin (A+B-C-D)+\sin (A+B-C+D)+\sin (A+B+C-D)=4 \sin (A+B) \cos C \cos D $$

$$ \cos 70^{\circ}-\cos 10^{\circ}+\sin 40^{\circ}=0 $$

$$ \sin 50^{\circ}-\sin 70^{\circ}+\sin 10^{\circ}=0 $$

$$ \sin 10^{\circ}+\sin 20^{\circ}+\sin 40^{\circ}+\sin 50^{\circ}=\sin 70^{\circ}+\sin 80^{\circ} $$

$$ \cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}=0 $$

$$ \cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ}=\cos 24^{\circ}+\cos 48^{\circ} $$

$$ \left[\sin 55^{\circ}-\sin 19^{\circ}\right]+\left[\sin 53^{\circ}-\sin 17^{\circ}\right]=\cos 1^{\circ} $$