Question

In Problems 47–72, establish each identity.

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

Short answer

The given identity is established.

Step-by-step solution

Step 1. Given Information 

We are given an identity,

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

Step 2. Establishing the identity  

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

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

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

Hence, the given expression is an identity.