Sin^3x + cos^3x = 1 + sinx.cosx

To prove the equation sin^3x + cos^3x = 1 + sinx*cosx, we can start by factoring the left side of the equation.

sin^3x + cos^3x = (sinx + cosx)(sin^2x - sinxcosx + cos^2x)

Now, we can use the Pythagorean identity sin^2x + cos^2x = 1 to simplify the expression further.

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

Now, let's focus on the right side of the equation.

1 + sinx*cosx = 1 + sinxcosx

We can see that the right side of the equation matches the simplified expression on the left side.

Therefore, sin^3x + cos^3x = 1 + sinx*cosx is true.