Question

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:

\(4\cos x - 4\cos9x\)