In the expansion of `((x)/(costheta)+(1)/(xsintheta))^(16)`, if `l_(1)` is the least value of the term independent of `x` when `(pi)/(8) le theta le (pi)/(4)` and `l_(2)` is the least value of the term independent of `x` when `(pi)/(16) le theta le (pi)/(8)`, then the value of `(l_(2))/(l_(1))` is

Options

A. `8`
B. `32`
C. `16`
D. `64`

Answer

C

Explanation

`(c )` General term `T_(r+1)=^(16)C_(r )((x)/(costheta))^(16-r)((1)/(xsintheta))^(r )`
`=^(16)C_(r )(1)/((costheta)^(16-r)(sintheta)^(r ))*x^(16-2r)`
If this term is independent of `x`, then `16-2r=0`
`:.` The term independent of `x=^(16)C_(8)(1)/(cos^(8)thetasin^(8)theta)`
`=^(16)C_(8)(2^(8))/(sin^(8)2theta)`
`l_(1)=^(16)C_(8)(2^(8))/(sin^(8)"(pi)/(2))=^(16)C_(8)2^(8)`
`l_(2)=^(16)C_(8)(2^(8))/(sin^(8)"(pi)/(4))=^(16)C_(8)*(2^(8))/(((1)/(sqrt(2)))^(8))=^(16)C_(8)2^(12)`
`:.(l_(2))/(l_(1))=(2^(12))/(2^(8))=2^(4)=16`