Question

How do you verify the identity $\sin \left(\frac{\pi }{2}+x\right)=\cos \left(x\right)$?

Answer 1

Verify the given expression:

It is known that $\sin \left(A+B\right)=\sin \left(A\right)·\cos \left(B\right)+\cos \left(A\right)·\sin \left(B\right)$.

Thus, $\sin \left(\frac{\pi }{2}+x\right)=\sin \left(\frac{\pi }{2}\right)·\cos \left(x\right)+\cos \left(\frac{\pi }{2}\right)·\sin \left(x\right)$

$$\begin{gathered} \Rightarrow \sin \left(\frac{\mathrm{\pi }}{2}+x\right)=\left(1\right)·\cos x+\left(0\right)·\sin x \\ \Rightarrow \sin \left(\frac{\mathrm{\pi }}{2}+x\right)=\cos x+0 \\ \Rightarrow \sin \left(\frac{\mathrm{\pi }}{2}+x\right)=\cos x \end{gathered}$$

Therefore, it is verified that $\sin (\frac{\mathrm{\pi }}{2}+\mathrm{x})=\cos x$.