Question

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

Answer 1

`\cos\left(x+\frac\pi2\right)=\cos x\cos\frac\pi2-\sin x\sin\frac\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+(\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=(\pi)/(2)`

Hence,

`\cos (x+(\pi)/(2))=\cos x\cos ((\pi)/(2))-\sin x\sin ((\pi)/(2))`

We know that:

`\cos (\pi)/(2)=0\\\\and\\\\\sin (\pi)/(2)=1`

Hence, we have:

`\cos (x+(\pi)/(2))=0-\sin x\\\\i.e.\\\\\cos (x+(\pi)/(2))=-\sin x`