Chapter 9.7, Problem 15E

To determine

Simplify the expression.

Answer to Problem 15E

The simplified form of expression $\frac{\cos \left(\frac{\pi }{2}-x\right)}{\csc x}$ is ${\sin }^{2}x$ .

Explanation of Solution

Given information:

The expression:

  $\frac{\cos \left(\frac{\pi }{2}-x\right)}{\csc x}$

Formula used:

  $\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta$

  $\frac{1}{\csc \theta }=\sin \theta$

  $\frac{\cos \left(\frac{\pi }{2}-x\right)}{\csc x}$

Applying the identity $\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta$ ,

  $\begin{array}{l}=\frac{\sin x}{\csc x}\\ \\ =\sin \theta \cdot \frac{1}{\csc \theta }\end{array}$

Applying the identity $\frac{1}{\csc \theta }=\sin \theta$ ,

  $\begin{array}{l}=\sin x\cdot \sin x\\ \\ ={\sin }^{2}x\end{array}$

Cancelling out ${\cos }^2\theta$

  $={\sin }^2\theta$

The simplified form of expression $\frac{\cos \left(\frac{\pi }{2}-x\right)}{\csc x}$ is ${\sin }^{2}x$ .