Express all trigonometric ratios in terms of $\cos A$.

Prove the following identity:

$\frac{1}{\sec q-1}-\frac{1}{\sec q+1}=2 {\cot }^2 q$

Prove the following identity:

${\sec }^2 q+{\mathrm{cosec}}^2 q=\sec q·\mathrm{cosec} q$

Prove the following identity:

${\sin }^{4}A+{\cos }^{4}A=1-2{\sin }^{2}A·{\cos }^{2}A$

Prove the following identity:

$\sqrt{\frac{1-\cos q}{1+\cos q}}=\cos ecq-cotq$

Prove the following identity:

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

Prove the following identity:

$\frac{1+\cos \theta }{1-\cos \theta }-\frac{1-\cos \theta }{1+\cos \theta }=4cot\theta ·\cos ec\theta$

Prove the following identity:

$\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin {\displaystyle \theta }}=2\cos ec\theta$

If $\cos q-\sin q=\sqrt{2}\sin q$ then prove that $\cos q+\sin q=\sqrt{2}\cos q$.