Using Taylor series to evaluate $\lim_{x\to0} \frac{\sqrt{1+ x\sin(x)} - \sqrt{\cos(2x)}}{ \tan^2(x/2)}$
Evaluate the limit $\lim_{x \to 0}{\frac{\sqrt{1+x\sin{x}}-\sqrt{\cos{2x}}}{\tan^{2}{\frac{x}{2}}}}$