Prove the following trigonometric identities

$(1-\sin \theta +\cos \theta {)}^2=2(1+\cos \theta \left)\right(1-\sin \theta )$

Prove that $(cosec\theta -\cot \theta {)}^2=\frac{1-\cos \theta }{1+\cos \theta }$

Prove that $\frac{1+\cos \theta -{\sin }^2\theta }{\sin \theta (1+\cos \theta )}=\cot \theta$

Prove that $\frac{\sin \theta -2{\sin }^3\theta }{2{\cos }^3\theta -\cos \theta }=\tan \theta$

Prove that $\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }={\cos }^2\theta -{\sin }^2\theta$

Prove that $\frac{1-{\tan }^2\theta }{{\cot }^2\theta -1}={\tan }^2\theta$

Prove that $\frac{\tan \theta }{\sec \theta -1}+\frac{\tan \theta }{\sec \theta +1}=2cosec\theta$

Prove that $\frac{\cot \theta }{cosec\theta +1}+\frac{cosec\theta +1}{\cot \theta }=2\sec \theta$

Prove that $\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin \theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2{\sin }^2\theta -1}$