`tan(pi/4+1/2cos^-1x)+tan(pi/4-1/2cos^-1x)`, `x!=0` is equal to

Options

A. `x`
B. `2x`
C. `(2)/(x)`
D. none of these

Answer

C

Explanation

`tan ((pi)/(4) + (1)/(2) cos^(-1) x) + tan ((pi)/(4) - (1)/(2) cos^(-1) x)`
`= (1 + tan ((1)/(2) cos^(-1) x))/(1 - tan((1)/(2) cos^(-1) x)) + (1- tan ((1)/(2) cos^(-1) x))/(1 + tan ((1)/(2) cos^(-1) x))`
`= ((1 + tan ((1)/(2) cos^(-1) x))^(2)+ (1- tan ((1)/(2) cos^(-1) x))^(2))/(1 - tan^(2) ((1)/(2) cos^(-1) x))`
`= 2(1 + tan^(2) ((1)/(2) cos^(-1) x))/(1 - tan^(2) ((1)/(2) cos^(-1) x))`
`= (2)/(cos(cos^(-1) x)) = (2)/(x)`