$9 \sec^{2} A - 9 \tan^{2} A = \dots$

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

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

State whether the following is true or false. Justify your answer.
$\sin \theta = \cos \theta$ for all values of $\theta$.

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$(\operatorname{cosec} \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$

In triangle $ABC,$ right-angled at $B,$ if $\tan A = \frac{1}{\sqrt{3}},$ find the value of:
$(i)$ $\sin A \cos C + \cos A \sin C$
$(ii)$ $\cos A \cos C - \sin A \sin C$