Pre-calc question

Question

Verify: sin^3 𝑥−cos^3𝑥/sinx-cosx = 1 + sin 𝑥 cos 𝑥

what do i do with the third powers?

Answer 1

If you factor sin³x - cos³x, you get (sin x -cos x)(sin²x + sin x cos x + cos² x)

similar to factoring x³-y³

The sin x - cos x cancels with the denominator, leaving sin²x + sin x cos x + cos² x

sin² x + cos²x = 1

Therefore, replacing sin² x + cos²x with 1 leaves

1 + sin x cos x = 1 + sin x cos x

Answer 2

(sin³x -cos³x)/(sinx - cosx) = (sinx - cosx)(sin²x + sinxcosx + sin²x)/(sinx - cosx) = 1 + sinxcosx