Verify the Identity sin(x)^4-cos(x)^4=2sin(x)^2-1

Problem

sin^4(x)−cos^4(x)=2*sin^2(x)−1

Solution

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

2. Factor the expression using the difference of squares formula 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. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 to simplify the second factor.

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

4. Substitute the identity cos^2(x)=1−sin^2(x) into the remaining expression to write everything in terms of sine.

sin^2(x)−(1−sin^2(x))

5. Simplify the expression by distributing the negative sign and combining like terms.

sin^2(x)−1+sin^2(x)=2*sin^2(x)−1

Final Answer

sin^4(x)−cos^4(x)=2*sin^2(x)−1

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