Simplify (sin(A)^4-cos(A)^4)/(cos(A)-sin(A))

Problem

(sin^4(A)−cos^4(A))/(cos(A)−sin(A))

Solution

1. Factor the numerator as a difference of squares, where sin^4(A)=(sin^2(A))^2 and cos^4(A)=(cos^2(A))^2

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

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

(sin^2(A)−cos^2(A))*(1)=sin^2(A)−cos^2(A)

3. Factor the remaining expression sin^2(A)−cos^2(A) as a difference of squares again.

sin^2(A)−cos^2(A)=(sin(A)−cos(A))*(sin(A)+cos(A))

4. Substitute the factored numerator back into the original fraction.

((sin(A)−cos(A))*(sin(A)+cos(A)))/(cos(A)−sin(A))

5. Rewrite the denominator by factoring out a negative sign to match the numerator factor.

cos(A)−sin(A)=−(sin(A)−cos(A))

6. Cancel the common factor (sin(A)−cos(A)) from the numerator and the denominator.

((sin(A)−cos(A))*(sin(A)+cos(A)))/(−(sin(A)−cos(A)))=−(sin(A)+cos(A))

7. Distribute the negative sign to reach the final simplified form.

−(sin(A)+cos(A))=−sin(A)−cos(A)

Final Answer

(sin^4(A)−cos^4(A))/(cos(A)−sin(A))=−sin(A)−cos(A)

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