Question

Prove that $\sin \left[\frac{\pi }{2}-x\right]=\cos x.$

Answer 1

As we have $\sin \left[\frac{\pi }{2}-x\right]=\cos x.$

As it can be written as $\sin \left[\frac{\pi }{2}-x\right]=\cos \left[\frac{\pi }{2}-\left(\frac{\pi }{2}-x\right)\right]\left[As\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta \right]$

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

Hence Proved