The range of the value of the term independent of x in the expansion of

 is 

If  is an independent term, then  
${\mathrm{T}}_6={10}_{{\mathrm{C}}_5}{\left({\sin }^{-1}\mathrm{\alpha }\right)}^5{\left({\cos }^{-1}\mathrm{\alpha }\right)}^5={10}_{{\mathrm{C}}_5}{\left({\sin }^{-1}{\mathrm{\alpha cos}}^{-1}\mathrm{\alpha }\right)}^5$
$\mathrm{f}\left(\mathrm{\alpha }\right)={\sin }^{-1}\mathrm{\alpha }\cdot {\cos }^{-1}\mathrm{\alpha }={\sin }^{-1}\mathrm{\alpha }\left(\frac{\mathrm{\pi }}{2}-{\sin }^{-1}\mathrm{\alpha }\right)$
$=-1\left[{\left({\sin }^{-1}\mathrm{\alpha }-\frac{\mathrm{\pi }}{4}\right)}^2-\frac{{\mathrm{\pi }}^2}{16}\right]=\frac{{\mathrm{\pi }}^2}{16}-{\left({\sin }^{-1}\mathrm{\alpha }-\frac{\mathrm{\pi }}{4}\right)}^2\mathrm{Max}\text{ of }\mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{\pi }}^2}{16}$
  Min of  Required Range $=\left[\frac{{ }^{-10}{C}_{C_5}{\pi }^{10}}{2^5},\frac{{10}_{C_5}{\pi }^{10}}{2^{20}}\right]$