Answer
- Apply the Addition Formula for sine to the left side.
- Simplify.
- The left side would be equal with the right one; thus, proving the identity:
$$\sin\Big(x-\frac{\pi}{2}\Big)=-\cos x$$
Work Step by Step
*Addition Formula for sine:
$$\sin(A+B)=\sin A\cos B+\cos A\sin B$$
$$\sin\Big(x-\frac{\pi}{2}\Big)=-\cos x$$
*Consider the left side and apply Addition Formula here:
$$\sin\Big(x-\frac{\pi}{2}\Big)=\sin\Big[x+\Big(-\frac{\pi}{2}\Big)\Big]=\sin x\cos\Big(-\frac{\pi}{2}\Big)+\cos x\sin\Big(-\frac{\pi}{2}\Big)$$
$$\sin\Big(x-\frac{\pi}{2}\Big)=\sin x\times0+\cos x\times(-1)$$
$$\sin\Big(x-\frac{\pi}{2}\Big)=-\cos x$$
The identity has been proved.
