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

Evaluate the following:

$\frac{5 \cos ^{2} 60^{\circ}+4 \sec ^{2} 30^{\circ}-\tan ^{2} 45^{\circ}}{\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}}$

If $A , B$ and $C$ are interior angles of a triangle $ABC ,$ then show that

$\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$

Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of

$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$

$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$

Show that:

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

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

Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.