express the given product as a sum or difference containing only sines or cosines. cos(3x)cos(5x)
express the given product as a sum or difference containing only sines or cosines. cos(3x)cos(5x)
Answer
Explanation:
Step1: Use the product - to - sum formula
The formula for (\cos A\cos B=\frac{1}{2}[\cos(A + B)+\cos(A - B)]). Here (A = 3x) and (B=5x). [ \begin{align*} \cos(3x)\cos(5x)&=\frac{1}{2}[\cos(3x + 5x)+\cos(3x-5x)]\ \end{align*} ]
Step2: Simplify the expressions inside the cosine functions
Simplify (3x + 5x=8x) and (3x-5x=-2x). Since (\cos(-\theta)=\cos\theta), we have: [ \begin{align*} \frac{1}{2}[\cos(3x + 5x)+\cos(3x-5x)]&=\frac{1}{2}[\cos(8x)+\cos(2x)] \end{align*} ]
Answer:
(\frac{1}{2}\cos(8x)+\frac{1}{2}\cos(2x))