Question

write the expression as a sum or difference of trigonometric functions. 6 sin 2x sin 5x 6 sin 2x 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 \(A = 2x\) and \(B=5x\).
So, \(6\sin2x\sin5x=6\times\frac{1}{2}[\cos(2x - 5x)-\cos(2x + 5x)]\).

Step2: Simplify the expression

First, \(6\times\frac{1}{2}=3\).
Then, \(2x-5x=-3x\) and \(2x + 5x = 7x\).
Since \(\cos(-\alpha)=\cos\alpha\), the expression becomes \(3[\cos3x-\cos7x]=3\cos3x-3\cos7x\).

Answer:

\(3\cos3x - 3\cos7x\)