You have written the wrong expression, the correct one is (after applying L'Hospital rule) :
$\lim_{x \to 0} \frac{-\sin(x)-\frac{1}{1+x}+1}{2\sin x\cos x} = \lim_{x \to 0} \frac{-\sin(x)}{2\sin x \cos x} + \frac{\frac{-1 + 1 +x}{1+x}}{2\sin x \cos x}$
$=\lim_{x \to 0} \frac{-1}{2 \cos x} + \frac{\frac{x}{1+x}}{2\sin x \cos x}$
Yes there should be (1+x) in denominator as well, but that doesn't change the answer because that evaluates to 1 after applying limit.