Chapter 7.2, Problem 23E

To determine

To prove: the given co function identity.

Explanation of Solution

Given information:

An identity is given as,

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

Formula used:

Subtraction formula of cosine

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

And

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

Calculation:

Consider the given identity.

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

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

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

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