# How do you verify the identity (tan x + cos x) /( 1+sin x )= sec x?

Aug 12, 2015

$\frac{\tan x + \cos x}{1 + \sin x} \ne \sec x$

#### Explanation:

Assuming that $\frac{\tan x + \cos x}{1 + \sin x} = \sec x$, it needs to be true for all $x$

Taking $x = \frac{\pi}{4}$:

LHS: $\frac{\tan \left(\frac{\pi}{4}\right) + \cos \left(\frac{\pi}{4}\right)}{1 + \sin \left(\frac{\pi}{4}\right)} = \frac{1 + \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} = 1$

RHS: $\sec \left(\frac{\pi}{4}\right) = \sqrt{2}$

Aug 12, 2015

You can't because $\frac{\tan x + \cos x}{1 + \sin x} \ne \sec x$

#### Explanation:

Assuming that $\frac{\tan x + \cos x}{1 + \sin x} = \sec x$, it needs to be true for all $x$

Taking $x = \frac{\pi}{4}$:

LHS: $\frac{\tan \left(\frac{\pi}{4}\right) + \cos \left(\frac{\pi}{4}\right)}{1 + \sin \left(\frac{\pi}{4}\right)} = \frac{1 + \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} = 1$

RHS: $\sec \left(\frac{\pi}{4}\right) = \sqrt{2}$