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Chapter 5: Problem 26

Verify the identity. $$\sec x-\cos x=\sin x \tan x$$

Short Answer

Expert verified
This identity \( \sec x - \cos x = \sin x \tan x \) has been verified to be true by the detailed steps.

Step by step solution

01

Express in terms of sine and cosine

Expressing \(\sec x\), \(\tan x\), and \(\cos x\) in terms of sine and cosine, the equation becomes: \( \frac{1}{\cos x} - \cos x = \sin x \frac{\sin x}{\cos x} \)
02

Simplify the equation

Simplify the equation to have like terms, this gives: \( \frac{1-\cos^{2} x} {\cos x} = \sin^{2} x \)
03

Apply Pythagorean Identity

Apply Pythagorean Identity \(\sin^{2} x = 1 - \cos^{2} x\), this gives: \(\frac{1-\cos^{2} x}{\cos x} = 1 - \cos^{2} x \)
04

Simplify the equation

Now cancel out the like terms, and this gives: \(1-\cos^{2} x = 1 - \cos^{2} x\)
05

Equate the two sides of the equation

The left hand side equals to the right hand side, the identity has been verified or proven to be true.

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