Chapter 7.2, Problem 26E

To determine

To prove: the given identity.

Explanation of Solution

Given information:

An identity is given as,

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

Formula used:

Subtraction formula of cosine

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

Calculation:

Consider the given identity.

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

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

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

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