Show: $\displaystyle \frac{1 + cos 2x}{csc x sin 2x} = cos x$

What I have so far:

(and I'm probably wrong)

$\displaystyle \frac{1 + cos^2x - sin^2x}{csc x 2sinx cos x}$

then csc x turns to $\displaystyle \frac{1}{sin x}$ in the denominator...

$\displaystyle \frac{1 + cos^2x - sin^2x}{\frac{2sin x cos x}{sin x}}$

cancels out one of the sin x...

$\displaystyle \frac{1 + cos^2x - sin^2x}{sin x cos x}$

then in the numerator I know I should use the "$\displaystyle cos^2x + sin^2x = 1$" rule somehow... but I can't figure out what to do next.

I appreciate your help!