Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 49

$$\cos(180^\circ+\theta)=-\cos\theta$$

$$A=\cos(180^\circ+\theta)$$ The strategy is to apply the cosine sum identity: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ That turns $A$ into $$A=\cos180^\circ\cos\theta-\sin180^\circ\sin\theta$$ We have $\cos180^\circ=-\cos0^\circ=-1$ and $\sin180^\circ=\sin0^\circ=0$ $$A=-1\times\cos\theta-0\times\sin\theta$$ $$A=-\cos\theta$$