The maximum value of `(cosalpha_(1))(cosalpha_(2))...(cosalpha_(n)),` under the restrications `0lealpha_(1),alpha_(2)..., alpha_(n)le(pi)/(2),and (cotalpha_(1))(cotalpha_(2))...(cotalpha_(n))=1is`

Options

A. `(1)/(2^(n)//2)`
B. `(1)/(2^(n))`
C. `(2)/(2n)`
D. 1

Answer

A

Explanation

We have,
`cot alpha_(1)cot alpha_(2)... cot alpha_(n)=1`
`impliescos alpha_(1)cosalpha_(2)... cos alpha_(n)=sinalpha_(1). sinalpha_(2)....sinalpha_(n)`
`implies(cosalphacosalpha_(2)...c osalpha_(n))^(n)`
`=(cosalpha_(1)sinalpha_(1))(cosalpha_(2)sinalpha_(2))...(cosalpha_(n)sinalpha_(n))`
`implies (cosalpha_(1)cosalpha_(2)...cosalpha_(n))^(2)`
`=(1)/(2^(n))(sin2 alpha_(1))(sin 2 alpha_(2))...(sin2 alpha_(n))`
`=(cos alpha_(1)cos alpha_(2)...cosalpha_(n))^(2)le (1)/(2^(n))`
`impliescos alpha_(1)cos alpha_(2)...cos alpha_(n)le(1)/(2^(n//2))`
Thus, the maximum value of cos `alpha_(1).cos alpha_(2)... cos alpha_(2)is (1)/(2^(n//2)).`