This follows directly from the double angle identity for cosine and the Pythagorean identity:
[tex]\cos2\theta=\cos^2\theta-\sin^2\theta=\begin{cases}2\cos^2\theta-1\\1-2\sin^2\theta\end{cases}[/tex]
So we have
[tex]\dfrac{1-2\sin^2\theta}{2\cos^2\theta-1}=\dfrac{\cos2\theta}{\cos2\theta}=1[/tex]
