Chapter 5.1, Problem 4E

To determine

To fill in the blank to complete the trigonometric identity $\csc \left(\frac{\pi }{2}-u\right)=\_\_\_\_$ .

Answer to Problem 4E

  $\sec u$

Explanation of Solution

Given:

The expressions, $\csc \left(\frac{\pi }{2}-u\right)=\_\_\_\_$

Concept Used:

Cofunction Identities:The value of any trigonometric function of an angle $\theta$ is equal to the cofunction of the complement of $\theta$ . The cofunction identities given by

  $\begin{array}{l}\sin \theta =\cos \left(\frac{\pi }{2}-\theta \right),\sin \theta =\cos \left(\frac{\pi }{2}-\theta \right)\\ \tan \theta =\cot \left(\frac{\pi }{2}-\theta \right),\cot \theta =\tan \left(\frac{\pi }{2}-\theta \right)\\ \sec \theta =\csc \left(\frac{\pi }{2}-\theta \right),\csc \theta =\sec \left(\frac{\pi }{2}-\theta \right)\end{array}$

Calculation:

In order tofind the equivalent expression for the given trigonometric identity $\csc \left(\frac{\pi }{2}-u\right)$ , use the cofunction identity of trigonometric functions.

Observe that the cofunction of cosecant (csc) function is secant (sec). So, using the cofunction identities, the given expression can be simplified as

  $\csc \left(\frac{\pi }{2}-u\right)=\sec u$

Thus, the complete trigonometric identity is $\csc \left(\frac{\pi }{2}-u\right)=\sec u$ .