Evaluate the following:
$2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$

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

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

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

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 $B$,$AB = 5 \, cm$ and $\angle ACB = 30^{\circ}$. Determine the lengths of the sides $BC$ and $AC$.