Prove the following identities :

$\dfrac{1}{\text{1 + cos A}} + \dfrac{1}{\text{1 - cos A}}$ = 2 cosec² A

Solving L.H.S. of the equation :

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

By formula,

1 - cos² A = sin² A

$\Rightarrow \dfrac{2}{\text{sin}^2 A} \\[1em] \Rightarrow 2\text{ cosec}^2 A.$

Since, L.H.S. = R.H.S.

Hence, proved that $\dfrac{1}{\text{1 + cos A}} + \dfrac{1}{\text{1 - cos A}}$ = 2 cosec² A.