Precalculus (6th Edition) Blitzer

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

Chapter 5 - Mid-Chapter Check Point - Page 684: 10


See the explanation below.

Work Step by Step

The given expression on the left side $\frac{1+\csc x}{\sec x}-\cot x$ can be further simplified by using the reciprocal identity $\csc x=\frac{1}{\sin x}$, $\sec x=\frac{1}{\cos x}$ and the quotient identity $\cot x=\frac{\cos x}{\sin x}$: $\begin{align} & \frac{1+\csc x}{\sec x}-\cot x=\frac{1+\frac{1}{\sin x}}{\frac{1}{\cos x}}-\frac{\cos x}{\sin x} \\ & =\cos x.\left( 1+\frac{1}{\sin x} \right)-\frac{\cos x}{\sin x} \\ & =\cos x+\frac{\cos x}{\sin x}-\frac{\cos x}{\sin x} \\ & =\cos x \end{align}$ Hence, the left side is equal to the right side $\frac{1+\csc x}{\sec x}-\cot x=\cos x$
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