Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$

If $\sec 4 A =\operatorname{cosec}\left( A -20^{\circ}\right),$ where $4 A$ is an acute angle, find the value of $A$. (in $^{\circ}$)

State whether the following are true or false. Justify your answer.

The value of $\sin \theta$ increases as $\theta$ increases.

Evaluate:

$\frac{\sin ^{2} 63^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\cos ^{2} 73^{\circ}}$

Evaluate $\frac{\tan 65^{\circ}}{\cot 25^{\circ}}$