$\frac{1+\tan ^{2} A}{1+\cot ^{2} A}=........$

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

The value of $\cos \theta$ increases as $\theta$ increases

If $\angle B$ and $\angle Q$ are acute angles such that $\sin B =\sin Q$, then prove that $\angle B =\angle Q$.

Evaluate the following:

$\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}$

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

$\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta$

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

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