Sum the series: `sqrt(1+cos alpha)+sqrt(1+cos 2alpha)+sqrt(1+cos 3alpha)` +.......to n terms, where `0ltalphaltpi`

Answer

`sqrt(2)(sin((nalpha)/(4)))/(sin((alpha)/(4)))cos[(n+1)(alpha)/(4)]`

Explanation

`S=sqrt(1+cosalpha)+sqrt(1+cos 2alpha)+sqrt(1+cos3alpha)+........n` terms
`=sqrt(2cos^(2)""(alpha)/(2))+sqrt(2cos^(2)alpha)+sqrt(2cos^(2)(3alpha)/(2))+........n ` terms
`=sqrt(2)[cos(alpha)/(2)+cos alpha+cos(3alpha)/(2)+.......+"to" n "terms"]`
`sqrt(2)(sin((nalpha)/(4)))/(sin((alpha)/(4)))cos(((alpha)/(2))+((alpha)/(2))+(n-1)((alpha)/(2)))/(2)`
`=sqrt(2)((sin((n alpha)/(4)))/(4))/(sin((alpha)/(4)))cos[(n+1)(alpha)/(4)]`