Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

$\dfrac{\text{cos A}}{\text{1 - sin A}} + \dfrac{\text{cos A}}{\text{1 + sin A}} = \text{2 sec A}.$

Solving L.H.S.,

$\Rightarrow \dfrac{\text{cos A(1 + sin A) + \text{cos A(1 - sin A)}}}{\text{(1 - sin A)(1 + sin A)}} \\[1em] = \dfrac{\text{cos A + cos A sin A + cos A - cos A sin A}}{1 - \text{sin}^2 A} \\[1em] = \dfrac{2\text{ cos A}}{\text{cos}^2 A} \\[1em] = \dfrac{2}{\text{cos A}} \\[1em] = 2\text{ sec A}.$

Since, L.H.S. = R.H.S. hence, proved that $\dfrac{\text{cos A}}{\text{1 - sin A}} + \dfrac{\text{cos A}}{\text{1 + sin A}} = 2\text{ sec A}$.