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

In a right triangle $ABC$,right-angled at $B$. If $\tan A = 1$,then verify that $2 \sin A \cos A = 1$.

In $\triangle PQR$,right-angled at $Q$,$PQ = 3 \, cm$ and $PR = 6 \, cm$. Determine $\angle QPR$ and $\angle PRQ$.

Evaluate $\frac{\tan 65^{\circ}}{\cot 25^{\circ}}$

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

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