$\underset{x\to {{(\frac{\pi }{2})}^{+}}}{\mathop{\lim }}\,\frac{\tan x}{\sqrt{1+\cos 2x}}=\frac{-\infty }{\sqrt{1+{{(-1)}^{+}}}}=\frac{-\infty }{\sqrt{{{0}^{+}}}}=\frac{-\infty }{{{0}^{+}}}=-\infty $