Evaluate the following:

$2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$

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

$\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta$

In $\triangle$ $ABC,$ right-angled at $B$, $AB =5\, cm$ and $\angle ACB =30^{\circ}$ (see $Fig.$). Determine the lengths of the sides $BC$ and $AC .$

In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:

$(i)$ $\sin A \cos C+\cos A \sin C$

$(ii)$ $\cos A \cos C-\sin A \sin C$

$\frac{2 \tan 30^{\circ}}{1+\tan ^{2} 30^{\circ}}=$

$\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}}=$