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

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

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

$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$

$(ii)$ cot $A$ is the product of cot and $A$.

$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.

In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:

$(i)$ $\sin A, \cos A$

$(ii)$ $\sin C, \cos C$

Show that:

$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ}=1$

$(ii)$ $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}=0$

Evaluate the following:

$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

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