Prove that $\sqrt{\frac{1+\sin \theta }{1-\sin \theta }}+\sqrt{\frac{1-\sin \theta }{1+\sin \theta }}=2\sec \theta$

Prove that $\sqrt{\frac{1+\cos \theta }{1-\cos \theta }}+\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=2\mathrm{cosec}\theta$

Prove that $\frac{1}{\sec \theta -\tan \theta }-\frac{1}{\cos {\displaystyle \theta }}=\frac{{\displaystyle 1}}{{\displaystyle \cos \theta }}-\frac{{\displaystyle 1}}{{\displaystyle \sec \theta +\tan \theta }}$

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

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

Prove that $\left(\sec \theta +\tan \theta -1\right)\left(\sec \theta -\tan \theta +1\right)=2\tan \theta$

Prove the following trigonometric identities: 

$\left(1+\cot \theta -\mathrm{cosec}\theta \right)\left(1+\tan \theta +\sec \theta \right)=2$

Prove that ${\sec }^6\theta ={\tan }^6\theta +3{\tan }^2\theta {\sec }^2\theta +1$

Prove that ${\mathrm{cosec}}^6\theta ={\cot }^6\theta +3{\cot }^2\theta {\mathrm{cosec}}^2\theta +1$