I'm not sure how to prove that the modulus of -cos2x in the range 0<x<pi/2 is less than 1.
You should know that
$\displaystyle -1 \leq -\cos{2x} \leq 1$
so
$\displaystyle |-\cos{2x}| \leq 1$.
You should also find the values for which $\displaystyle -\cos{2x} = \pm 1$, to show that they occur outside the domain $\displaystyle 0 < x < \frac{\pi}{2}$.