On the left side, multiply numerator and denominator by [tex]\cos^2\theta[/tex]:
[tex]\dfrac{\sec^2\theta-1}{\sec^2\theta}=\dfrac{\cos^2\theta(\sec^2\theta-1)}{\cos^2\theta\sec^2\theta}=\dfrac{1-\cos^2\theta}1[/tex]
which follows from the fact that [tex]\sec\theta=\dfrac1{\cos\theta}[/tex]. Then apply the Pythagorean identity,
[tex]\sin^2\theta+\cos^2\theta=1\implies\dfrac{\sec^2\theta-1}{\sec^2\theta}=\sin^2\theta[/tex].
and we're done.
