Find the greatest value of the term independent of $x$ in the expansion of ${(x sin α + \frac{cos α}{x})}^{10}$ , where $α \in R$

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Solution

Term independent of x: $^{10} C_{5} (x sin α)^{5} \cdot {(\frac{cos α}{x})}^{5}$
$=^{10} C_{5} (sin α cos α)^{5}$
$=^{10} C_{5} {(\frac{sin 2 α}{2})}^{5}$
Maximum value of $sin 2 α = 1$ :
$∴ {[^{10} C_{5} {(\frac{sin 2 α}{2})}^{5}]}_{max}^{10} =^{10} C_{5} \frac{1}{2^{5}}$
$= \frac{^{10} C_{5}}{32}$

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