Prove that: (cos^3x - sin^3x)/(cosx - sinx) = 1 + cosx*sinx is always true.

We have to prove that [(cos x)^3 - (sin x)^3]/[cos x - sin
x]= 1 + cos x * sin x

Starting with the left hand
side:

(cos x)^3 - (sin x)^3 / cos x - sin
x

use a^3 - b^3 = (a - b)(a^2 + ab +
b^2)

=> [(cos x - sin x)[(cos x)^2 + cos x * sin x +
(sin x)^2]]/( cos x - sin x)

cancel (cos x - sin
x)

=> [(cos x)^2 + cos x * sin x + (sin
x)^2]

Use (cos x)^2 + (sin x)^2 =
1

=> 1 + cos x * sin
x

This is the right hand
side.

This proves that [(cos x)^3 - (sin
x)^3]/[cos x - sin x] = 1 + cos x * sin x