Verify if sin^3 x - cos^3 x = (sinx - cosx)(1 + sinxcosx)

We'll have to verify if the expression from the left side
is equal to the expression from the right side.

We'll start
by re-writting the difference of the cubes from the left side. We'll use the
formula:

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

We'll substitute a and b by sin x and cos x and we'll
get:

(sin x)^3 -  (cos x)^3 = (sin x - cos x)[(sin x)^2 +
sin x*cos x + (cos x)^2]

But the sum (sin x)^2 + (cos x)^2
= 1, from the fundamental formula of trigonometry.

We'll
substitute the sum of squares by the value
1.

(sin x)^3 -  (cos x)^3 = (sin x - cos x)(1
+ sin x*cos x) q.e.d