Question

In Problems47–72, establish each identity.

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

Short answer

The given identity is established.

Step-by-step solution

Step 1. Given Information 

We are given an identity,

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

Step 2. Establishing the identity  

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

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

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

The given identity is established.