Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 48

$$\cos(90^\circ+\theta)=-\sin\theta$$

$$A=\cos(90^\circ+\theta)$$ Here we need to use the cosine sum identity: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ Applying that identity to $A$: $$A=\cos90^\circ\cos\theta-\sin90^\circ\sin\theta$$ $$A=0\times\cos\theta-1\times\sin\theta$$ $$A=-\sin\theta$$