Express the product as a sum containing only sines or cosines.
-$\sin ( 4 \theta ) \sin ( 6 \theta )$
A) $\sin ^ { 2 } \left( 24 \theta ^ { 2 } \right)$
B) $\frac { 1 } { 2 } [ - \cos ( 2 \theta ) - \cos ( 10 \theta ) ]$
C) $\frac { 1 } { 2 } [ \cos ( 2 \theta ) - \cos ( 10 \theta ) ]$
D) $\frac { 1 } { 2 } [ \cos ( 10 \theta ) - \sin ( 2 \theta ) ]$

Question

Quizplus · Accepted Answer

The product-to-sum formula for sine functions is $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$. Applying this to $\sin(4\theta)\sin(6\theta)$, we get $\frac{1}{2}[\cos(4\theta-6\theta) - \cos(4\theta+6\theta)] = \frac{1}{2}[\cos(-2\theta) - \cos(10\theta)]$. Since $\cos(-2\theta) = \cos(2\theta)$, the correct expression is $\frac{1}{2}[\cos(2\theta) - \cos(10\theta)]$.

Express the Product as a Sum Containing Only Sines or Cosines