use the product - to - sum identities to rewrite the following expression as a sum or difference…

Explanation:

Step1: Recall product - to - sum identity

The product - to - sum identity for $\sin A\sin B$ is $\sin A\sin B=\frac{1}{2}[\cos(A - B)-\cos(A + B)]$. Here $A = 3x$ and $B=4x$.

Step2: Apply the identity

We have $4\sin(3x)\sin(4x)=4\times\frac{1}{2}[\cos(3x - 4x)-\cos(3x + 4x)]$.

Step3: Simplify the expression

$4\times\frac{1}{2}[\cos(3x - 4x)-\cos(3x + 4x)] = 2[\cos(-x)-\cos(7x)]$. Since $\cos(-x)=\cos x$, the expression becomes $2[\cos x-\cos(7x)]=2\cos x-2\cos(7x)$.

Answer:

$2\cos x - 2\cos(7x)$