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

$(\sec A + \tan A)(1 - \sin A) = \dots$

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

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$

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

If $\tan (A + B) = \sqrt{3}$ and $\tan (A - B) = \frac{1}{\sqrt{3}}$,where $0^{\circ} < A + B \leq 90^{\circ}$ and $A > B$,find the values of $A$ and $B$.