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