Question

Prove the identity.

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

Short answer

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

Step-by-step solution

Step 1. Given information.

The given equation, $\sin \left(\frac{\pi }{2}-x\right)=\sin \left(\frac{\pi }{2}+x\right)$.

Step 2. Proving the identities using the other identities.

Taking the left-hand side:

$=\sin \frac{\pi }{2}\cos x-\cos \frac{\pi }{2}\sin x$

Putting the values:

$=\cos x$

Taking the right-hand side:

$=\sin \frac{\pi }{2}\cos x+\cos \frac{\pi }{2}\sin x$

Putting the values:

$=\cos x$

Both sides give the same result, hence the identity is proved.