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