If the value of $\underset{x\to 0}{\lim } (2-\cos \mathrm{x}\sqrt{\cos 2\mathrm{x}}{)}^{\left(\frac{\mathrm{x}+2}{{\mathrm{x}}^2}\right)}$ is $e^{-\alpha }$ then $\alpha =$ 

$$\begin{gathered} \underset{x\to 0}{\lim }(2-\cos x\sqrt{\cos 2x}{)}^{\frac{x+2}{x^2}} \\ ={\mathrm{e}}^{\underset{x\to 0}{\lim }\left[\frac{\mathrm{x}+2}{{\mathrm{x}}^2}\right][2-\cos \mathrm{x}\sqrt{\cos 2\mathrm{x}}-1]} \\ =e^{\frac{\mathrm{Lt}}{x\to 0}}(1-\cos x\sqrt{\cos 2x})\cdot \left(\frac{x+2}{x^2}\right) \\ \mathrm{ }\boxed{\cos 2x=1-\frac{4x^2}{2!}+\frac{16x^4}{4!}--- \\ \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots \dots \\ } \\ \cos x\sqrt{\cos 2x}=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}--\right){\left(1-\frac{4x^2}{2!}+\frac{16x^4}{4!}---\right)}^{1/2}\because dr. degree is x^2 \end{gathered}$$We have to extract till the coefficient of ${\mathrm{x}}^2$ in numerator   $e^{\underset{x\to 0}{\lim }\left[1-\left(1-\frac{{\mathrm{x}}^2}{2!}\right)(1-2{\mathrm{x}}^2{)}^{1/2}\right]\frac{\mathrm{x}+2}{{\mathrm{x}}^2}}$

${\mathrm{e}}^{\underset{\mathrm{x}\to 0}{\lim }\left|1-(1-\frac{{\mathrm{x}}^2}{2})(1-x^2\right|\frac{\mathrm{x}+2}{{\mathrm{x}}^2}} {\mathrm{e}}^{\underset{\mathrm{x}\to 0}{\lim }\frac{3}{2}{\mathrm{x}}^2\left(\frac{\mathrm{x}+2}{{\mathrm{x}}^2}\right)}$