Question

Prove that:

`(sin^3 theta + cos^3 theta)/(sin theta + cos theta) = 1 - sin theta cos theta`

Answer 1

LHS = `(sin^3 theta + cos^3 theta)/(sin theta + cos theta)`

`= ((sin theta + cos theta)(sin^2 theta - sin theta cos theta + cos^2 theta))/(sin theta + cos theta)`     ...[∵ a³ + b³ = (a + b)(a² - ab + b²)]

= sin² θ - sin θ cos θ + cos² θ

= 1 - sin θ cos θ      ...[sin² θ + cos² θ = 1]

= RHS  

Hence Proved.