Question

write the expression as a sum or difference of trigonometric functions. 4 sin 2x sin 6x 4 sin 2x sin 6x =

Explanation:

Step1: Use the product - to - sum formula

The product - to - sum formula for \(\sin A\sin B\) is \(\sin A\sin B=\frac{1}{2}[\cos(A - B)-\cos(A + B)]\). Here \(A = 2x\) and \(B=6x\).
So, \(4\sin2x\sin6x=4\times\frac{1}{2}[\cos(2x - 6x)-\cos(2x + 6x)]\)

Step2: Simplify the expression

First, simplify \(4\times\frac{1}{2}=2\).
Then, simplify the arguments of the cosine functions: \(2x-6x=-4x\) and \(2x + 6x=8x\).
Since \(\cos(-\alpha)=\cos\alpha\), the expression becomes \(2[\cos4x-\cos8x]=2\cos4x-2\cos8x\)

Answer:

\(2\cos4x - 2\cos8x\)