# Prove the following. 1 1 − sin θ + 1 1 + sin θ = 2 sec 2 θ - Geometry

#### Question

Prove the following.

$\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta} = 2 \sec^2 \theta$

#### Solution

$\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta}$

$= \frac{1 + \sin\theta + 1 - \sin\theta}{\left( 1 - \sin\theta \right)\left( 1 + \sin\theta \right)}$

$= \frac{2}{1 - \sin^2 \theta} \left[ \left( a - b \right)\left( a + b \right) = a^2 - b^2 \right]$

$= \frac{2}{\cos^2 \theta} \left( \sin^2 \theta + \cos^2 \theta = 1 \right)$

$= 2 \sec^2 \theta$

Is there an error in this question or solution?