Simplify (sin^3 beta + cos^3 beta)/(sin beta +cos beta)

I'll use 'x' instead of beta since its easier to type and gets less messy:

((sin(x))^3 + (cos(x))^3) / (sin(x) + cos(x))
factor the top using this formula -------> a^3 + b^3 = (a + b)(a^2 - ab + b^2)
= (sin(x) + cos(x))((sin(x))^2 - sin(x)*cos(x) + (cos(x))^2) / (sin(x) + cos(x))

the (sin(x) + cos(x)) in the numerator and denominator cancels out:
= (sin(x))^2 - sin(x)*cos(x) + (cos(x))^2

use this trig identity -----> (sin(x))^2 + (cos(x))^2 = 1
= 1 - sin(x)cos(x)

thats it