Find the following limit problem I'm trying to find the following limit 
$$ \lim_{x\to 0} \left({1-\sin x } {\cos x}\right)^{{ \csc 2x} }$$
How to prove the above limit equals $e^{-{1\over2}}$?

$$ \lim_{x\to 0} \left(1-\sin x \cos x\right)^{\csc 2x}=\lim_{x\to 0} \left(1-\sin x \cos x\right)^{\frac{1}{\sin 2x}}=$$
$$ \lim_{x\to 0} \left(1-\sin x \cos x\right)^{\frac{1}{\sin x\cos x}\frac{1}{2}}=e^{-1\cdot\frac{1}{2}}$$
