Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.1 - Verifying Trigonometric Identities - Exercise Set - Page 658: 12

Answer

See the explanation below.

Work Step by Step

To verify the given identity, $\tan \theta +\cot \theta =\sec \theta \csc \theta $ Recall Trigonometric Identities, $\begin{align} & {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\ & \tan \theta =\frac{\sin \theta }{\cos \theta } \\ & \cot \theta =\frac{\cos \theta }{\sin \theta } \\ \end{align}$ Use the above identities and solve the left side of the given expression, $\begin{align} & \tan \theta +\cot \theta =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta } \\ & =\frac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta }{\sin \theta \cos \theta } \\ & =\frac{1}{\sin \theta \cos \theta } \\ & =\frac{1}{\sin \theta }\left( \frac{1}{\cos \theta } \right) \end{align}$ Recall Reciprocal Identities, $\begin{align} & \csc \theta =\frac{1}{\sin \theta } \\ & \sec \theta =\frac{1}{\cos \theta } \\ \end{align}$ Therefore, $\tan \theta +\cot \theta =\sec \theta \csc \theta $ Hence, it is proved that the given identity holds true.
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.