Question

Verify the identity. cos quantity x plus pi divided by two = -sin x

Answer 1

$\cos\left(x+\dfrac\pi2\right)=\cos x\cos\dfrac\pi2-\sin x\sin\dfrac\pi2=0\cos x-1\sin x=-\sin x$

via the angle sum identity for cosine.

Answer 2

Answer with explanation:

We are asked to prove the identity:

         $\cos (x+\dfrac{\pi}{2})=-\sin x$

We know that:

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

Here we have:

$A=x\ and\ B=\dfrac{\pi}{2}$

Hence,

$\cos (x+\dfrac{\pi}{2})=\cos x\cos (\dfrac{\pi}{2})-\sin x\sin (\dfrac{\pi}{2})$

We know that:

$\cos \dfrac{\pi}{2}=0\\\\and\\\\\sin \dfrac{\pi}{2}=1$

Hence, we have:

$\cos (x+\dfrac{\pi}{2})=0-\sin x\\\\i.e.\\\\\cos (x+\dfrac{\pi}{2})=-\sin x$