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

Problem

sin4(x)−cos4(x)=2*sin2(x)−1

Solution

1. Identify the left side of the equation 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. Substitute the identity cos2(x)=1−sin2(x) into the remaining expression to write everything in terms of sine.

sin2(x)−(1−sin2(x))

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

sin2(x)−1+sin2(x)=2*sin2(x)−1

Final Answer

sin4(x)−cos4(x)=2*sin2(x)−1

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