Answer
- Apply the Addition Formula for sine for the left side.
- Simplify.
- Then the left side would be equal with the right one.
Thus, the identity will be proved: $$\sin\Big(x+\frac{\pi}{2}\Big)=\cos x$$
Work Step by Step
*Addition Formulas 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:
$$\sin\Big(x+\frac{\pi}{2}\Big)=\sin x\cos\Big(\frac{\pi}{2}\Big)+\cos x\sin\Big(\frac{\pi}{2}\Big)$$
$$\sin\Big(x+\frac{\pi}{2}\Big)=\sin x\times0+\cos x\times1$$
$$\sin\Big(x+\frac{\pi}{2}\Big)=\cos x$$
The identity has been proved.
