Prove the following:

$\frac{1}{\tan 3A-\tan A}-\frac{1}{\cot 3A-\cot A}=\cot 2A$

Prove the following:

$\cos 7°\cos 14°\cos 28°\cos 56°=\frac{\sin 68°}{16\cos 83°}$

Prove the following:

Prove the following: 

Prove the following:

${\cos }^{2}x+{\cos }^2(x+120°)+{\cos }^2(x-120°)=\frac{3}{2}$

Prove the following: 

$2\mathrm{cosec}2x+\mathrm{cosec}x=\sec x\cot \left(\frac{x}{2}\right)$

Prove the following:

$4\cos x\cos \left(x+\frac{\pi }{3}\right)\cos \left(x-\frac{\pi }{3}\right)=\cos 3x$

Prove the following: 

$\sin x\tan \left(\frac{x}{2}\right)+2\cos x=\frac{2}{1+{\tan }^2\left({\displaystyle \frac{x}{2}}\right)}$

Express the following as a sum or difference of two trigonometric function.

$2\sin 4x\cos 2x$