Verify the identity: cos x + sin x tan x = sec x.


Verify the identity: {eq}cos x + sin x tan x = sec x. {/eq}


We have to prove that both sides of the given equation are equal. This is done by taking the more complicated side (the L.H.S. here) and trying to simplify it till we make it the same as the other side.

Answer and Explanation:

The identity is verified as follows.

$$\begin{align} \cos x + \sin x \tan x &= \sec x\\ \cos x + \sin x \frac{\sin x}{\cos x} &= \sec x\\ \cos x + \frac{\sin^2 x}{\cos x} &= \sec x\\ \frac{\cos^2 x+\sin^2 x}{\cos x} &= \sec x\\ \frac{1}{\cos x} &= \sec x&&&&\left [ \because \sin ^2x+\cos^2x=1 \right ]\\ \therefore \sec x=\sec x \end{align} $$

Learn more about this topic:

Trigonometric Identities: Definition & Uses

from Honors Precalculus Textbook

Chapter 23 / Lesson 1

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