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

Problem

(sin4(x)−cos4(x))/(sin2(x)−cos2(x))

Solution

1. Identify the numerator as a difference of squares in the form a2−b2 where a=sin2(x) and b=cos2(x)

2. Factor the numerator 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. Substitute the factored expression back into the original fraction.

((sin2(x)−cos2(x))*(sin2(x)+cos2(x)))/(sin2(x)−cos2(x))

4. Cancel the common factor (sin2(x)−cos2(x)) from the numerator and the denominator.

sin2(x)+cos2(x)

5. Apply the Pythagorean identity sin2(x)+cos2(x)=1

1

Final Answer

(sin4(x)−cos4(x))/(sin2(x)−cos2(x))=1

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