If $\sin 2A=\lambda \sin 2B$, then prove that $\frac{\tan (A+B)}{\tan (A-B)}=\frac{\lambda +1}{\lambda -1}$.

Prove the following:

$\frac{2\cos 2A+1}{2\cos A-1}=\tan \left({60}^{\circ }+A\right)\tan \left({60}^{\circ }-A\right)$

Prove the following:

$\tan A+\tan (60°+A)+\tan (120°+A)=3\tan 3A$

Prove the following:

$3{\tan }^{6}10°-27{\tan }^{4}10°+33{\tan }^{2}10°=1$

Prove the following.

$\mathrm{cosec}{48}^{\circ }+\mathrm{cosec}{96}^{\circ }+\mathrm{cosec}{192}^{\circ }+\mathrm{cosec}{384}^{\circ }=0$

Prove the following:

$3(\sin x-\cos x{)}^4 + 6(\sin x+\cos x{)}^2 +4({\sin }^{6}x+{\cos }^{6}x)=13$

Prove the following:

$\tan A+2\tan 2A+4\tan 4A+8\cot 8A=\cot A$

If $A+B+C=\frac{3\mathrm{\pi }}{2}$, then prove that $\cos 2A+\cos 2B+\cos 3C=1-4\sin A\sin B\sin C$.

In any triangle $ABC, \sin A-\cos B=\cos C$ prove that$\angle B=\frac{\pi }{2}$.