Use the sum-to-product formulas to find the exact value of the given expression. $\cos 150 ^ { \circ } - \cos 30 ^ { \circ }$
A) 1
B) $- 1$
C) $- \sqrt { 3 }$
D) $- \frac { \sqrt { 3 } } { 2 }$
E) 0

Question

Quizplus · Accepted Answer

The sum-to-product formula for cosine is given by $\cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)$. Applying this to $\cos 150^\circ - \cos 30^\circ$, we get $-2 \sin \left( \frac{150^\circ + 30^\circ}{2} \right) \sin \left( \frac{150^\circ - 30^\circ}{2} \right) = -2 \sin 90^\circ \sin 60^\circ = -2 \cdot 1 \cdot \frac{\sqrt{3}}{2} = -\sqrt{3}$.

Use the Sum-To-Product Formulas to Find the Exact Value of the Given