The greatest value of the term independent of x, as $\alpha$varies over R , in the expansion of ${\left(x\cos \alpha +\frac{\sin \alpha }{x}\right)}^{20} is$

For Independent terms

       $\mathrm{r}=\frac{\mathrm{np}}{\mathrm{p}+\mathrm{q}}=\frac{\left(20\right) \left(1\right)}{1+1}=10$

$\therefore$ Independent term $={}^{20}{\mathrm{C}}_{10} {(\cos \mathrm{\alpha })}^{10} {(\sin \mathrm{\alpha })}^{10}$

                                       $={}^{20}{\mathrm{C}}_6 \frac{\left(\sin 2\mathrm{\alpha }\right)}{2^{10}}$

            Greatest Value$={}^{20}{\mathrm{C}}_{10} {\left(\frac{1}{2}\right)}^{10}$