verify that the equation given below is an identity. cos(π/2 + x)= - sin x rewrite the expression on the…

verify that the equation given below is an identity. cos(π/2 + x)= - sin x rewrite the expression on the left using an appropriate identity. cos(π/2 + x)=cos π/2 cos x - sin π/2 sin x apply the sum identity for cosine. cos(π/2 + x)=□ - □ simplify both of the terms obtained in the previous step.

verify that the equation given below is an identity. cos(π/2 + x)= - sin x rewrite the expression on the left using an appropriate identity. cos(π/2 + x)=cos π/2 cos x - sin π/2 sin x apply the sum identity for cosine. cos(π/2 + x)=□ - □ simplify both of the terms obtained in the previous step.

Answer

Explanation:

Step1: Apply cosine - sum identity

The sum identity for cosine is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here $A=\frac{\pi}{2}$ and $B = x$, so $\cos(\frac{\pi}{2}+x)=\cos\frac{\pi}{2}\cos x-\sin\frac{\pi}{2}\sin x$.

Step2: Evaluate trigonometric values

We know that $\cos\frac{\pi}{2} = 0$ and $\sin\frac{\pi}{2}=1$. Substituting these values into the right - hand side of the equation from Step 1, we get $0\times\cos x-1\times\sin x$.

Step3: Simplify the expression

$0\times\cos x-1\times\sin x=0 - \sin x=-\sin x$.

Answer:

Since the left - hand side $\cos(\frac{\pi}{2}+x)$ simplifies to $-\sin x$, which is the right - hand side of the given equation, the equation $\cos(\frac{\pi}{2}+x)=-\sin x$ is an identity.