Simplify sin(8x)-sin(2x)

Problem

sin(8*x)−sin(2*x)

Solution

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

2. Apply the formula sin(A)−sin(B)=2*cos((A+B)/2)*sin((A−B)/2) where A=8*x and B=2*x

3. Substitute the values into the formula:

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

4. Simplify the arguments inside the trigonometric functions:

sin(8*x)−sin(2*x)=2*cos((10*x)/2)*sin((6*x)/2)

5. Divide the terms to reach the final simplified form:

sin(8*x)−sin(2*x)=2*cos(5*x)*sin(3*x)

Final Answer

sin(8*x)−sin(2*x)=2*cos(5*x)*sin(3*x)

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