Simplify (sin(x)^4-cos(x)^4)/(sin(x)^2-cos(x)^2)

Problem

(sin^4(x)−cos^4(x))/(sin^2(x)−cos^2(x))

Solution

1. Identify the numerator 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 numerator 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. Substitute the factored expression back into the original fraction.

((sin^2(x)−cos^2(x))*(sin^2(x)+cos^2(x)))/(sin^2(x)−cos^2(x))

4. Cancel the common factor (sin^2(x)−cos^2(x)) from the numerator and the denominator.

sin^2(x)+cos^2(x)

5. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1

1

Final Answer

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

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