Simplify sin(x)^4-cos(x)^4

Problem

sin4(x)−cos4(x)

Solution

1. Identify the expression as a difference of squares in the form a2−b2 where a=sin2(x) and b=cos2(x)

2. Factor the expression using the difference of squares formula a2−b2=(a−b)*(a+b)

sin4(x)−cos4(x)=(sin2(x)−cos2(x))*(sin2(x)+cos2(x))

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the second factor.

sin4(x)−cos4(x)=(sin2(x)−cos2(x))*(1)

4. Apply the double angle identity for cosine, cos(2*x)=cos2(x)−sin2(x) which implies sin2(x)−cos2(x)=−cos(2*x)

sin4(x)−cos4(x)=−cos(2*x)

Final Answer

sin4(x)−cos4(x)=−cos(2*x)

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