Prove the trigonometric identity $\frac{\cot\left(\infty \right)^2+1}{\cot\left(\infty \right)\cdot \sec\left(\infty \right)^2}=\cot\left(\infty \right)$

Prove the identity

$tan\left(x\right)+cot\left(x\right)=sec\left(x\right)cosec\left(x\right)$

Prove the identity

$\frac{1+\cos2x}{\sin2x}=\cot x$

Prove the identity

$sec\left(x\right)-tan\left(x\right)sin\left(x\right)=\frac{1}{sec\left(x\right)}$

Prove the identity

$\tan\left(x\right)+\cot\left(x\right)=\sec\left(x\right)\csc\left(x\right)$

Prove the identity

$\tan\left(a\right)+\cot\left(a\right)=\sec\left(a\right)\csc\left(a\right)$

Prove the identity

$\sec x-\sin x\tan x=\cos x$

Prove the identity

$\frac{1-sin\left(x\right)}{cos\left(x\right)}=\frac{cos\left(x\right)}{1+sin\left(x\right)}$