Chapter 9.7, Problem 19E

To determine

Simplify the expression.

Answer to Problem 19E

The simplified form of expression $\frac{\cos \left(\frac{\pi }{2}-\theta \right)}{\csc \theta }+{\cos }^2\theta$ is $1$ .

Explanation of Solution

Given information:

The expression:

  $\frac{\cos \left(\frac{\pi }{2}-\theta \right)}{\csc \theta }+{\cos }^2\theta$

Formula used:

  ${\sin }^2\theta +{\cos }^2\theta =1$

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

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

  $\frac{\cos \left(\frac{\pi }{2}-\theta \right)}{\csc \theta }+{\cos }^2\theta$

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

  $\begin{array}{l}=\frac{\sin \theta }{\csc \theta }+{\cos }^2\theta \\ \\ =\sin \theta \cdot \frac{1}{\csc \theta }+{\cos }^2\theta \end{array}$

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

  $\begin{array}{l}=\sin \theta \cdot \sin \theta +{\cos }^2\theta \\ \\ ={\sin }^2\theta +{\cos }^2\theta \end{array}$

Applying the identity ${\sin }^2\theta +{\cos }^2\theta =1$ ,

  $=1$

The simplified form of expression $\frac{\cos \left(\frac{\pi }{2}-\theta \right)}{\csc \theta }+{\cos }^2\theta$ is $1$ .