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

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