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

State whether the following are true or false. Justify your answer.
$(i)$ The value of $\tan A$ is always less than $1$.
$(ii)$ $\sec A = \frac{12}{5}$ for some value of angle $A$.

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

Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.

Given $15 \cot A = 8$,find $\sin A$ and $\sec A$.

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{1+\sec A}{\sec A} = \frac{\sin^2 A}{1-\cos A}$