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

Question

Quizplus · Accepted Answer

The product-to-sum formula for cosines states that $\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$. Applying this formula with $A = 2\theta$ and $B = 4\theta$, we get $\frac{1}{2}[\cos(2\theta - 4\theta) + \cos(2\theta + 4\theta)] = \frac{1}{2}[\cos(-2\theta) + \cos(6\theta)]$. Since $\cos(-\theta) = \cos(\theta)$, the expression simplifies to $\frac{1}{2}[\cos(2\theta) + \cos(6\theta)]$.

Express the Product as a Sum Containing Only Sines or Cosines