How do you prove csc2x = 1/2(secx)(cscx)?

$\csc 2 x = \frac{1}{\sin 2 x} = \frac{1}{2 \sin x . \cos x} = \left(\frac{1}{2}\right) . \left(\frac{1}{\sin} x\right) . \left(\frac{1}{\cos} x\right)$=
= $\left(\frac{1}{2}\right) . \left(\csc x\right) . \left(\sec x\right)$