Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 50

$$\cos(270^\circ+\theta)=\sin\theta$$

$$A=\cos(270^\circ+\theta)$$ The strategy is to apply the cosine sum identity: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ That turns $A$ into $$A=\cos270^\circ\cos\theta-\sin270^\circ\sin\theta$$ We have $\cos270^\circ=\cos(-90^\circ)=\cos90^\circ=0$ and $\sin270^\circ=\sin(-90^\circ)=-\sin90^\circ=-1$ $$A=0\times\cos\theta-(-1)\times\sin\theta$$ $$A=1\times\sin\theta=\sin\theta$$