Chapter 7.2, Problem 25E

To determine

To prove: the given identity.

Explanation of Solution

Given information:

An identity is given as,

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

Formula used:

Subtraction formula of sine

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

Calculation:

Consider the given identity.

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

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

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

Hence, the identity $\sin \left(x-\frac{\pi }{2}\right)=-\cos x$ is verified.