Question

How do you simplify $\cos \left(x-\frac{\mathrm{\pi }}{2}\right)$

Answer 1

Solve for the required value

We know that $\cos$ is an even function

$\Rightarrow \cos (-x)=\cos x$

So, for the given function,

$$\begin{gathered} \cos \left(x-\frac{\mathrm{\pi }}{2}\right)=\cos \{-\left(\frac{\mathrm{\pi }}{2}–x\right)\}=\cos \left(\frac{\mathrm{\pi }}{2}–x\right) \\ \Rightarrow \cos \left(x-\frac{\mathrm{\pi }}{2}\right)=\sin x \end{gathered}$$

Hence, the required answer is $\cos \left(x-\frac{\mathrm{\pi }}{2}\right)=\sin x$