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: 41

Answer

$$\cot\theta+\tan\theta=\sec\theta\csc\theta$$ The identity is proved to be true.

Work Step by Step

$$\cot\theta+\tan\theta=\sec\theta\csc\theta$$ We would try simplifying the left side first, using the identities $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$ $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ The left side then would be $$\cot\theta+\tan\theta$$ $$=\frac{\cos\theta}{\sin\theta}+\frac{\sin\theta}{\cos\theta}$$ $$=\frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}$$ $$=\frac{1}{\sin\theta\cos\theta}$$ (for $\sin^2\theta+\cos^2\theta=1$) $$=\frac{1}{\sin\theta}\times\frac{1}{\cos\theta}$$ $$=\sec\theta\csc\theta$$ (we know that $\sec\theta=\frac{1}{\cos\theta}$ and $\csc\theta=\frac{1}{\sin\theta}$) The identity is therefore proved.
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.