Chapter 7.2, Problem 24E

To determine

To prove: the given co function identity.

Explanation of Solution

Given information:

An identity is given as,

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

Formula used:

Subtraction formula of sine

  $\sin \left(u-v\right)=\sin u\cos v-\cos u\sin v$

And

  $\begin{array}{l}\sec x=\frac{1}{\cos x}\\ \sin x=\frac{1}{\sin x}\end{array}$

Calculation:

Consider the given identity.

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

Now, take left hand side of the identity and use the subtraction formula as shown:

  $\begin{array}{c}\csc \left(\frac{\pi }{2}-u\right)=\frac{1}{\sin \left(\frac{\pi }{2}-u\right)}\\ =\frac{1}{\sin \frac{\pi }{2}\cos u-\cos \frac{\pi }{2}\sin u}\\ =\frac{1}{1\times \cos u-0\times \sin u}\\ =\frac{1}{\cos u}\\ =\sec u\end{array}$

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