Prove the identity: $\;$ $\sin A \left(1 + \tan A\right) + \cos A \left(1 + \cot A\right) = \sec A + \text{cosec }A$
$\begin{aligned}
LHS & = \sin A \left(1 + \tan A\right) + \cos A \left(1 + \cot A\right) \\\\
& = \sin A \left(1 + \dfrac{\sin A}{\cos A}\right) + \cos A \left(1 + \dfrac{\cos A}{\sin A}\right) \\\\
& = \sin A \left(\dfrac{\sin A + \cos A}{\cos A}\right) + \cos A \left(\dfrac{\sin A + \cos A}{\sin A}\right) \\\\
& = \left(\sin A + \cos A\right) \left(\dfrac{\sin A}{\cos A} + \dfrac{\cos A}{\sin A}\right) \\\\
& = \left(\sin A + \cos A\right) \left(\dfrac{\sin^2 A + \cos^2 A}{\sin A \; \cos A}\right) \\\\
& = \dfrac{\sin A + \cos A}{\sin A \; \cos A} \;\;\; \left[\text{Note: } \sin^2 A + \cos^2 A = 1\right] \\\\
& = \dfrac{\sin A}{\sin A \; \cos A} + \dfrac{\cos A}{\sin A \; \cos A} \\\\
& = \dfrac{1}{\cos A} + \dfrac{1}{\sin A} \\\\
& = \sec A + \text{cosec }A \\\\
& = RHS
\end{aligned}$
Hence proved.