write the expression as a sum or difference of trigonometric functions. 8 sin 4x sin 5x 8 sin 4x sin 5x =

Explanation:

Step1: Use the product - to - sum formula

The formula for (\sin A\sin B=\frac{1}{2}[\cos(A - B)-\cos(A + B)]). Here (A = 4x) and (B=5x). So, (8\sin4x\sin5x=8\times\frac{1}{2}[\cos(4x - 5x)-\cos(4x + 5x)])

Step2: Simplify the expression

First, simplify (8\times\frac{1}{2}=4). Then, (\cos(4x - 5x)=\cos(-x)) and (\cos(4x + 5x)=\cos9x). Since (\cos(-\theta)=\cos\theta) (cosine is an even function), (\cos(-x)=\cos x). So, (4[\cos x-\cos9x]=4\cos x-4\cos9x)

Answer:

(4\cos x - 4\cos9x)