Prove that  $(1+\sec x)(cosecx-\cot x)\equiv \tan x$.

Prove that  $\frac{1}{\tan x+\cot x}=\sin x·\cos x$.

Prove that ${\sec }^{2}x+\sec x·\tan x\equiv \frac{1}{1-\sin x}$.

Prove that $\frac{1-{\cos }^{2}x}{{\sec }^{2}x-1}=1-{\sin }^{2}x$.

Prove that $\frac{1+{\tan }^2 x}{\tan x}=\sec x·cosecx$.

Prove that $\frac{\sin x}{1-{\cos }^2 x}=cosecx$.

Prove that $\frac{1+\sin x}{1-\sin x}=(\tan x+\sec x{)}^2$.

Prove that $\frac{1}{1+\cos x}+\frac{1}{1-\cos x}\equiv 2{cosec}^{2}x$.

Prove that $\frac{\cos x}{1+\sin x}+\frac{\cos x}{1-\sin x}\equiv 2\sec x$.