In the given figure,find $\tan P - \cot R$.

In $\triangle OPQ$,right-angled at $P$,$OP = 7\, cm$ and $OQ - PQ = 1\, cm$. Determine the values of $\sin Q$ and $\cos Q$.

Evaluate the following:
$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

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

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 $A$.

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$(\operatorname{cosec} \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$