The greatest value of the term independent (Coefficient of $x^0$) of x in the expansion of ${\left(xSin\alpha +\frac{Cos\alpha }{x}\right)}^8,\alpha \in R$ is

$\begin{array}{r}{T}_{r+1}{=}^8{C}_r(x\mathrm{Sin}\alpha {)}^{8-r}{\left(\frac{\mathrm{Cos}\alpha }{x}\right)}^r\\ {=}^8{C}_r{x}^{8-2r}{\mathrm{Sin}}^{8-r}\alpha {\mathrm{Cos}}^r\alpha \\ \text{ Independent term means }8-2r=0\Rightarrow 8=2r\Rightarrow r=4\\ \therefore T_{4+1}=8_{C_4}{\mathrm{Sin}}^4\alpha {\mathrm{Cos}}^4\alpha \\ =\left(\frac{8_{c_4}}{2^4}\right){\mathrm{Sin}}^4\left(2\alpha \right)\\ \text{ Greatest value }=\frac{8_{c_4}}{2^4}\left(\because \text{ Greatest value of }{\mathrm{Sin}}^4\left(2\alpha \right)=1\right.\text{ ) }\end{array}$