Prove the following trigonometric identity:

$\frac{1+\cos x}{1-\cos x}=\frac{{\tan }^{2}x}{(\sec x-1{)}^2}$

Prove the following trigonometric identity:

$\frac{{\tan }^{2}x}{{\tan }^{2}x-1}+\frac{{cosec}^{2}x}{{\sec }^{2}x-{cosec}^{2}x}=\frac{1}{{\sin }^{2}x-{\cos }^{2}x}$

Prove the following trigonometric identity:

${\sin }^{4}x-{\cos }^{4}x=1-2{\cos }^{2}x$

Prove the following trigonometric identity:

${\sec }^{4}x-{\tan }^{4}x=1+2{\tan }^{2}x$

Prove the following trigonometric identity:

${\sin }^{6}x+{\cos }^{6}x=1-3{\sin }^{2}x{\cos }^{2}x$

Prove the following trigonometric identity:

$\frac{1+\tan x}{\sin x}+\frac{1+\cot x}{\cos x}=2(\sec x+cosecx)$

Prove the following trigonometric identity:

$(\sin x+cosecx{)}^2+(\cos x+\sec x{)}^2={\tan }^{2}x+{\cot }^{2}x+7$

Prove the following trigonometric identity:

${cosec}^{6}x-{\cot }^{6}x=3{\cot }^{2}x{cosec}^{2}x+1$

Prove the following trigonometric identity:

${\sec }^{6}x-{\tan }^{6}x=1+3{\tan }^{2}x+3{\tan }^{4}x$