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

- Apply Addition Formula for cosine to the left side. - Simplify. Then both sides would be equal, proving the identity: $$\cos\Big(x+\frac{\pi}{2}\Big)=-\sin x$$
*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 and apply Addition Formula here: $$\cos\Big(x+\frac{\pi}{2}\Big)=\cos x\cos\Big(\frac{\pi}{2}\Big)-\sin x\sin\Big(\frac{\pi}{2}\Big)$$ $$\cos\Big(x+\frac{\pi}{2}\Big)=\cos x\times0-\sin x\times1$$ $$\cos\Big(x+\frac{\pi}{2}\Big)=-\sin x$$ The identity has been proved.