Factor sin(x)^3-cos(x)^3

Problem

sin^3(x)−cos^3(x)

Solution

1. Identify the expression as a difference of cubes, which follows the algebraic pattern a^3−b^3=(a−b)*(a^2+a*b+b^2)

2. Assign the terms to the variables in the formula by letting a=sin(x) and b=cos(x)

3. Substitute these values into the difference of cubes formula to get (sin(x)−cos(x))*(sin^2(x)+sin(x)*cos(x)+cos^2(x))

4. Simplify the second factor using the Pythagorean identity sin^2(x)+cos^2(x)=1

Final Answer

sin^3(x)−cos^3(x)=(sin(x)−cos(x))*(1+sin(x)*cos(x))

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