Answer
See below.
Work Step by Step
$LHS=\frac{sin^3\theta+cos^3\theta}{sin\theta+cos\theta}=\frac{(sin\theta+cos\theta)(sin^2\theta-sin\theta cos\theta+cos^2\theta)}{sin\theta+cos\theta}=sin^2\theta+cos^2\theta-sin\theta cos\theta=1-sin\theta cos\theta=RHS$
