Verify the Identity cos(x)^4-sin(x)^4=cos(2x)

Problem

cos^4(x)−sin^4(x)=cos(2*x)

Solution

1. Identify the left side of the equation as a difference of squares, where a^2−b^2=(a−b)*(a+b)

2. Factor the expression cos^4(x)−sin^4(x) by treating it as (cos^2(x))^2−(sin^2(x))^2

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

3. Apply the Pythagorean identity cos^2(x)+sin^2(x)=1 to simplify the second factor.

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

4. Recognize the double angle identity for cosine, which states that cos(2*x)=cos^2(x)−sin^2(x)

cos^4(x)−sin^4(x)=cos(2*x)

Final Answer

cos^4(x)−sin^4(x)=cos(2*x)

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