Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 202: 11

Answer

$$\frac{1}{1+\cos x}-\frac{1}{1-\cos x}=-2\cos x\csc^2 x$$

Work Step by Step

$$A=\frac{1}{1+\cos x}-\frac{1}{1-\cos x}$$ $$A=\frac{1-\cos x}{(1+\cos x)(1-\cos x)}-\frac{1+\cos x}{(1+\cos x)(1-\cos x)}$$ $$A=\frac{1-\cos x-(1+\cos x)}{(1+\cos x)(1-\cos x)}$$ $$A=\frac{1-\cos x-1-\cos x}{(1+\cos x)(1-\cos x)}$$ $$A=\frac{-2\cos x}{(1+\cos x)(1-\cos x)}$$ $$A=\frac{-2\cos x}{1-\cos^2 x}$$ (for $a^2-b^2=(a-b)(a+b)$) - Pythagorean Identity: $$\sin^2 x=1-\cos^2 x$$ Therefore, $$A=\frac{-2\cos x}{\sin^2 x}$$ $$A=-2\cos x(\frac{1}{\sin x})^2$$ - Reciprocal Identity: $$\csc x=\frac{1}{\sin x}$$ So, $$A=-2\cos x\csc^2 x$$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.