Prove the trigonometric identity $\cos\left(x-y\right)+\cos\left(x+y\right)=2\cos\left(x\right)\cos\left(y\right)$

Prove the identity

$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

Prove the identity

$\frac{\csc\left(x\right)}{\cot\left(x\right)}=\sec\left(x\right)$

Prove the identity

$\tan\left(x\right)\cdot \cos\left(x\right)\cdot \csc\left(x\right)=1$

Prove the identity

$\sin\left(x\right)^2+\cos\left(x\right)^2=1$

Prove the identity

$\csc\left(x\right)\cdot\tan\left(x\right)=\sec\left(x\right)$

Prove the identity

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

Prove the identity

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