Answer
- Apply Addition Formula for cosine to the left side.
- Simplify.
Then both sides would be equal, proving the identity:
$$\cos\Big(x+\frac{\pi}{2}\Big)=-\sin x$$
Work Step by Step
*Addition Formulas for cosine:
$$\cos(A+B)=\cos A\cos B-\sin A\sin B$$
$$\cos\Big(x+\frac{\pi}{2}\Big)=-\sin x$$
*Consider the left side and apply Addition Formula here:
$$\cos\Big(x+\frac{\pi}{2}\Big)=\cos x\cos\Big(\frac{\pi}{2}\Big)-\sin x\sin\Big(\frac{\pi}{2}\Big)$$
$$\cos\Big(x+\frac{\pi}{2}\Big)=\cos x\times0-\sin x\times1$$
$$\cos\Big(x+\frac{\pi}{2}\Big)=-\sin x$$
The identity has been proved.
