Simplify cos(8x)-cos(6x)

Problem

cos(8*x)−cos(6*x)

Solution

1. Identify the appropriate sum-to-product identity for the difference of two cosines.

cos(A)−cos(B)=−2*sin((A+B)/2)*sin((A−B)/2)

2. Assign the values for the variables in the identity based on the given expression.

A=8*x

B=6*x

3. Substitute the values into the sum-to-product formula.

cos(8*x)−cos(6*x)=−2*sin((8*x+6*x)/2)*sin((8*x−6*x)/2)

4. Simplify the arguments inside the sine functions.

(8*x+6*x)/2=(14*x)/2=7*x

(8*x−6*x)/2=(2*x)/2=x

5. Combine the simplified terms to reach the final product form.

−2*sin(7*x)*sin(x)

Final Answer

cos(8*x)−cos(6*x)=−2*sin(7*x)*sin(x)

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