Precalculus (6th Edition) Blitzer

Published by Pearson

Chapter 5 - Section 5.1 - Verifying Trigonometric Identities - Exercise Set - Page 658: 14

Answer

See the explanation below.

Work Step by Step

To verify the given identity, $\frac{\cos \theta \sec \theta }{\cot \theta }=\tan \theta$ Recall Trigonometric Identities, \begin{align} & \sec \theta =\frac{1}{\cos \theta } \\ & \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} & \frac{\cos \theta \sec \theta }{\cot \theta }=\frac{\frac{\cos \theta }{1}\cdot \frac{1}{\cos \theta }}{\frac{\cos \theta }{\sin \theta }} \\ & =\frac{1}{\frac{\cos \theta }{\sin \theta }} \\ & =1\div \frac{\cos \theta }{\sin \theta } \\ & =1\cdot \frac{\sin \theta }{\cos \theta } \end{align} Therefore, $\frac{\cos \theta \sec \theta }{\cot \theta }=\tan \theta$ Hence, it is proved that the given identity holds true.

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