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