Chapter 5.3, Problem 41E

Solution

To prove: The reduction formula, $\csc \left(\frac{\pi }{2}-u\right)=\sec u$ .

Given information:

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

Formula used:

Use the formula of sine of difference.

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

Calculation:

Simplify the left hand side of the given identity by sine of a difference formula.

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

Simplify further.

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

Therefore, it is proved that $\csc \left(\frac{\pi }{2}-u\right)=\sec u$ .