Simplify cos(4x)-cos(2x)

Problem

cos(4*x)−cos(2*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=4*x

B=2*x

3. Substitute the values of A and B into the sum-to-product formula.

cos(4*x)−cos(2*x)=−2*sin((4*x+2*x)/2)*sin((4*x−2*x)/2)

4. Simplify the arguments inside the sine functions.

(4*x+2*x)/2=(6*x)/2=3*x

(4*x−2*x)/2=(2*x)/2=x

5. Combine the results to write the final simplified expression.

−2*sin(3*x)*sin(x)

Final Answer

cos(4*x)−cos(2*x)=−2*sin(3*x)*sin(x)

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