Question

In Problems 47–72, establish each identity.

$\sin (\pi -\theta )=\sin \theta$

Short answer

The given identity is established.

Step-by-step solution

Step 1. Given Information 

We are given an identity,

$\sin (\pi -\theta )=\sin \theta$

Step 2. Establishing the identity 

We will establish the given expression as an identity by simplifying one side to give the other side.

Using $\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta$, we get

$$\begin{gathered} \sin (\pi -\theta )=\sin \pi \cos \theta -\cos \pi \sin \theta \\ \sin (\pi -\theta )=0·\cos \theta -(-1)·\sin \theta \\ \sin (\pi -\theta )=\sin \theta \end{gathered}$$

The given identity is established.