Chapter 11

$$ \sin \frac{\theta}{2} \sin \frac{7 \theta}{2}+\sin \frac{3 \theta}{2} \sin \frac{11 \theta}{2}=\sin 2 \theta \sin 5 \theta \text { . } $$

$$ \cos 2 \theta \cos \frac{\theta}{2}-\cos 3 \theta \cos \frac{9 \theta}{2}=\sin 5 \theta \sin \frac{5 \theta}{2} \text { . } $$

$$ \sin A \sin (A+2 B)-\sin B \sin (B+2 A)=\sin (A-B) \sin (A+B) . $$

$$ (\sin 3 A+\sin A) \sin A+(\cos 3 A-\cos A) \cos A=0 $$

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

$$ \frac{\sin A \sin 2 A+\sin 3 A \sin 6 A+\sin 4 A \sin 13 A}{\sin A \cos 2 A+\sin 3 A \cos 6 A+\sin 4 A \cos 13 A}=\tan 9 A $$

$$ \frac{\cos 2 A \cos 3 A-\cos 2 A \cos 7 A+\cos A \cos 10 A}{\sin 4 A \sin 3 A-\sin 2 A \sin 5 A+\sin 4 A \sin 7 A}=\cot 6 A \cot 5 A $$

$$ \cos \left(36^{\circ}-A\right) \cos \left(36^{\circ}+A\right)+\cos \left(54^{\circ}+A\right) \cos \left(54^{\circ}-A\right)=\cos 2 A $$

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

$$ \sin \left(45^{\circ}+A\right) \sin \left(45^{\circ}-A\right)=\frac{1}{2} \cos 2 A $$

$$ \sin A \sin \left(A+120^{\circ}\right)+\sin \left(A+120^{\circ}\right) \sin \left(A+240^{\circ}\right)+\sin \left(A+240^{\circ}\right) \sin A=-\frac{3}{4} $$

$$ \sin (A-B) \cos (A+B)+\sin (B-C) \cos (B+C)+\sin (C-D) \cos (C+D)+\sin (D-A) \cos (D+A)=0 $$

$$ \sin (\beta-\gamma) \cos (a-\delta)+\sin (\gamma-a) \cos (\beta-\delta)+\sin (a-\beta) \cos (\gamma-\delta)=0 $$

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

$$ \sin A \sin B \sin (A-B)+\sin B \sin C \sin (B-C)+\sin C \sin A \sin (C-A)+\sin (A-B) \sin (B-C) \sin (C-A)=0 $$

$$ 2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}=0 $$

$$ \tan 70^{\circ}-\tan 20^{\circ}=2 \tan 50^{\circ} $$

$$ \cos 20^{\circ} \cos 100^{\circ}+\cos 100^{\circ} \cos 140^{\circ}-\cos 140^{\circ} \cos 200^{\circ}=-\frac{3}{4} $$

$$ \sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}=\frac{1}{8} $$

$$ \cos 36^{\circ} \cos 42^{\circ} \cos 78^{\circ}=\frac{1}{8} $$

$$ \sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}=\frac{3}{16} $$

$$ \sin 6^{\circ} \sin 42^{\circ} \sin 66^{\circ} \sin 78^{\circ}=\frac{1}{16} $$

$$ \cos 6^{\circ} \cos 42^{\circ} \cos 66^{\circ} \cos 78^{\circ}=\frac{1}{16} $$

$$ \tan 6^{\circ} \tan 42^{\circ} \tan 66^{\circ} \tan 78^{\circ}=1 $$

$$ \frac{\sin 2 A}{1+\cos 2 A}=\tan A $$

$$ \frac{\sin 2 A}{1-\cos 2 A}=\cot A $$

$$ \frac{1-\cos 2 A}{1+\cos 2 A}=\tan ^{2} A $$

$$ \tan A+\cot A=2 \operatorname{cosec} 2 A $$

$$ \tan A-\cot A=-2 \cot 2 A $$

$$ \operatorname{cosec} 2 A+\cot 2 A=\cot A $$

$$ \frac{\tan 5 \theta+\tan 3 \theta}{\tan 5 \theta-\tan 3 \theta}=4 \cos 2 \theta \cos 4 \theta $$

$$ \frac{\sec 8 A-1}{\sec 4 A-1}=\frac{\tan 8 A}{\tan 2 A} $$

$$ \frac{1+\tan ^{2}\left(45^{\circ}-A\right)}{1-\tan ^{2}\left(45^{\circ}-A\right)}=\operatorname{cosec} 2 A $$

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

$$ \tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta . $$

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

$$ \cot \left(A+15^{\circ}\right)-\tan \left(A-15^{\circ}\right)=\frac{4 \cos 2 A}{1+2 \sin 2 A} $$

$$ \frac{\sin \theta+\sin 2 \theta}{1+\cos \theta+\cos 2 \theta}=\tan \theta $$

$$ \sin (2 n+1) A \sin A=\sin ^{2}(n+1) A-\sin ^{2} n A $$

$$ \frac{\sin (A+3 B)+\sin (3 A+B)}{\sin 2 A+\sin 2 B}=2 \cos (A+B) \text { . } $$

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

$$ \tan 2 A=(\sec 2 A+1) \sqrt{\sec ^{2} A-1} $$

$$ \cos ^{3} 2 \theta+3 \cos 2 \theta=4\left(\cos ^{6} \theta-\sin ^{6} \theta\right) $$

$$ 1+\cos ^{2} 2 \theta=2\left(\cos ^{4} \theta+\sin ^{4} \theta\right) $$

$$ \sec ^{2} A(1+\sec 2 A)=2 \sec 2 A $$

$$ \operatorname{cosec} A-2 \cot 2 A \cos A=2 \sin A $$

$$ \cos 4 a=1-8 \cos ^{2} a+8 \cos ^{4} a $$

$$ \sin 4 A=4 \sin A \cos ^{3} A-4 \cos A \sin ^{3} A $$

$$ \tan 3 A \tan \angle A \tan A=\tan 3 A-\tan \angle A-\tan A $$

$$ \sqrt{2+\sqrt{2+2 \cos 4 \theta}}=2 \cos \theta $$