Find the greatest value of the term independent of `x` in the expansion of `(xsinalpha+(cosalpha)/x)^(10)` , where `alpha in R`.

Explanation

We have,
`(r+1)^(th)` term in the given expansion of `(xsinalpha+x^(-1)cosalpha)^(10)`
`=^(10)C_(r)(xsinalpha)^(10-r)(x^(-1)cosalpha)^(r)`
`=.^(10)C_(r)(sinalpha)^(10-r)(cosalpha)^(r)x^(10-2r)`
for inependent term of x, `10-2r=0impliesr=5`
hence the requried term independent of x.
`=.^(10)C_(5)(sinalphacosalpha)^(5)=.^(10)C_(5)*(1)/(2^(5))(sin2alpha)^(5)`
`le.^(10)C_(5).(1)/(2^(5))=(10!)/((5!)^(2))*(1)/(2^(5))`
therefore, the greatest value of term independent of x
`=(10!)/((5!)^(2))*(1)/(32)`