Expand the Trigonometric Expression sin(x)^4-cos(x)^4

Problem

sin^4(x)−cos^4(x)

Solution

1. Identify the expression as a difference of squares in the form a^2−b^2 where a=sin^2(x) and b=cos^2(x)

2. Apply the formula for the difference of squares, which is a^2−b^2=(a−b)*(a+b)

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

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

sin^4(x)−cos^4(x)=(sin^2(x)−cos^2(x))*(1)

4. Apply the double angle formula for cosine, noting that cos(2*x)=cos^2(x)−sin^2(x) which means sin^2(x)−cos^2(x)=−cos(2*x)

sin^4(x)−cos^4(x)=−cos(2*x)

Final Answer

sin^4(x)−cos^4(x)=−cos(2*x)

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