Expert: Sherman D. - 5/8/2008

Question
(sinA+cosA-sin^3A-cos^3A) = sinAcos^2A + sin^2cosA

Answer
sin(A) + cos(A) - sin(A)^3 - cos(A)^3 = sin(A)cos(A)^2 + cos(A)sin(A)^2

(sin(A) + cos(A)) - (sin(A)^3 + cos(A)^2) =

(sin(A) + cos(A)) - (sin(A) + cos(A))(sin(A)^2 - sin(A)cos(A) + cos(A)^2) =

(sin(A) + cos(A))(1 - (sin(A)^2 + cos(A)^2 - sin(A)cos(A)) =

(sin(A) + cos(A))(1 - (1 - sin(A)cos(A))) =

(sin(A) + cos(A))(1 - 1 + sin(A)cos(A)) =

(sin(A)cos(A))(sin(A) + cos(A))

cos(A)sin(A)^2 + sin(A)cos(A)^2 which is the same as your answer.

sometimes it easier to think of the problem like this

x + y - x^3 - y^3, this way its easier to understand how to factor it, keeping in mind that x is still sin(A) and y is still cos(A).

sometimes you will run into problems like sin(A)^2 + 2sin(A) + 1, in which this is like saying x^2 + 2x + 1, so you can use the quadratic formula, keeping in mind that x = sin(A).

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another way to have done your problem would be

sin(A) + cos(A) - sin(A)^3 - cos(A)^3 =
sin(A) - sin(A)^3 + cos(A) - cos(A)^3 =
sin(A)(1 - sin(A)^2) + cos(A)(1 - cos(A)^2) =
sin(A)cos(A)^2 + cos(A)sin(A)^2

If you like, you can go with this one, since its simplier to understand and write.