Answer
- Apply Addition Formula for cosine to the left side.
- Simplify.
Then, we have 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:
$$\cos\Big(x-\frac{\pi}{2}\Big)=\cos\Big[x+\Big(-\frac{\pi}{2}\Big)\Big]$$
Apply Addition Formulas here:
$$\cos\Big(x-\frac{\pi}{2}\Big)=\cos x\cos\Big(-\frac{\pi}{2}\Big)-\sin x\sin\Big(-\frac{\pi}{2}\Big)$$
$$\cos\Big(x-\frac{\pi}{2}\Big)=\cos x\times0-\sin x\times(-1)$$
$$\cos\Big(x-\frac{\pi}{2}\Big)=\sin x$$
The identity has been proved.
