# How do you prove cosX / (secX - tanX) = 1 + sinX?

Mar 11, 2018

As below.

#### Explanation:

To prove $\cos \frac{x}{\sec x - \tan x} = \left(1 + \sin x\right)$

L H S  = cos x / ((1/cos x) - (sin x / cos x) as color(blue)(sec x = 1/cos x, tan x = sin x / cos x

$\implies \cos \frac{x}{\frac{1 - \sin x}{\cos} x}$ as color(green)(cos x  is the L C M of Denominator.

$\implies {\cos}^{2} \frac{x}{1 - \sin x}$

$\implies = \frac{1 - {\sin}^{2} x}{1 - \sin x}$ as color(blue)(cos^2x = 1 - sin^2x

$\implies \frac{\left(1 + \sin x\right) \cdot \textcolor{red}{\cancel{1 - \sin x}}}{\textcolor{red}{\cancel{1 - \sin x}}}$

$\implies 1 + \sin x$

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