Evaluate:
$\frac{\sin 18^{\circ}}{\cos 72^{\circ}}$

Evaluate:
$\frac{\tan 26^{\circ}}{\cot 64^{\circ}}$

Consider $\triangle ACB$,right-angled at $C$,in which $AB = 29$ units,$BC = 21$ units and $\angle ABC = \theta$. Determine the values of:
$(i)$ $\cos^2 \theta + \sin^2 \theta$
$(ii)$ $\cos^2 \theta - \sin^2 \theta$

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

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 $\triangle OPQ$,right-angled at $P$,$OP = 7\, cm$ and $OQ - PQ = 1\, cm$. Determine the values of $\sin Q$ and $\cos Q$.