Evaluate $\frac{\tan 65^{\circ}}{\cot 25^{\circ}}$

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 $A$.

If $\tan (A + B) = \sqrt{3}$ and $\tan (A - B) = \frac{1}{\sqrt{3}}$,where $0^{\circ} < A + B \leq 90^{\circ}$ and $A > B$,find the values of $A$ and $B$.

Prove that $\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity $\sec ^{2} \theta=1+\tan ^{2} \theta.$

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

In a right triangle $ABC$,right-angled at $B$. If $\tan A = 1$,then verify that $2 \sin A \cos A = 1$.