# How do you prove 1/(1+sin) + 1/(1-sin) = 2sec^2 ?

Feb 5, 2016

see explanation

#### Explanation:

To prove trig identity either manipulate the left side into the form of the right side or the right side into the form of the left side or manipulate both sides together until they agree.

choose left side:

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

( distribute brackets on denominator)

$= \frac{2}{1 - {\sin}^{2} x}$

now $\left(1 - {\sin}^{2} x\right) = {\cos}^{2} x$

$\Rightarrow \frac{2}{1 - {\sin}^{2} x} = \frac{2}{{\cos}^{2} x} = 2 {\sec}^{2} x \textcolor{b l a c k}{\text{ = RHS }}$