Complete the sentence so the result is an identity. Let x be any real number.
-$2 \sin 8 x \sin 17 x$
A) $\sin 25 x + \sin 9 x$
B) $\frac { 1 } { 2 } ( \cos 25 x + \cos 9 x )$
C) $\cos 25 x + \cos 9 x$
D) $\cos 9 x - \cos 25 x$

Question

Quizplus · Accepted Answer

This is an application of the product-to-sum identities in trigonometry. Specifically, the identity for the product of sines is given by $\sin A \sin B = \frac{1}{2}(\cos(A-B) - \cos(A+B))$. Applying this to $2 \sin 8x \sin 17x$ gives $\cos(17x - 8x) - \cos(17x + 8x)$, which simplifies to $\cos 9x - \cos 25x$, matching option D.

Complete the Sentence So the Result Is an Identity 2sin⁡8xsin⁡17x2 \sin 8 x \sin 17 x2sin8xsin17x

Complete the Sentence So the Result Is an Identity $2 \sin 8 x \sin 17 x$