Chapter 7.2, Problem 27E

To determine

To prove: the given identity.

Explanation of Solution

Given information:

An identity is given as,

  $\sin \left(x-\pi \right)=-\sin x$

Formula used:

Subtraction formula of sine

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

Calculation:

Consider the given identity.

  $\sin \left(x-\pi \right)=-\sin x$

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

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

Hence, the identity $\sin \left(x-\pi \right)=-\sin x$ is verified.