Explanation

(d)
The general term in the expansion of $\left( xcos\alpha + \frac{sin\alpha}{x} \right)^{20}$ is ${\ ^{20}C_{r}\left( x\cos\alpha \right)}^{20 - r}\left( \frac{\sin\alpha}{x} \right)^{r} = \ ^{20}C_{r}x^{20 - 2r}{(\cos\alpha)}^{20 - r}{(\sin\alpha)}^{r}$
For this term to be independent of $x$, we get
$20 - 2r = 0\ \Rightarrow \ r = 10$
Let $\beta =$Term independent of $x$
$= \ \ ^{20}C_{10}{(\cos\alpha)}^{10}{(\sin\alpha)}^{10}$
$= \ ^{20}C_{10}{(\cos\alpha\sin\alpha)}^{10}$
$= \ ^{20}C_{10}\left( \frac{\sin{2a}}{2} \right)^{10}$
Thus, the greatest possible value of $\beta$ is $\ ^{20}C_{10}\left( \frac{1}{2} \right)^{10}$