## University Calculus: Early Transcendentals (3rd Edition)

- 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: $$\sin\Big(x+\frac{\pi}{2}\Big)=\cos x$$
*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.