If `cot^(-1)x+cot^(-1)y+cot^(-1)z=pi/2,x , y , z >0a n dx y<1,` then `x+y+z` is also equal to (a)`1/x+1/y+1/z` (b) `x y z` (c) `xy+yz+zx` (d) none of these

Options

A. `(1)/(x) + (1)/(y) + (1)/(z)`
B. `xyz`
C. `xy + yz + zx`
D. none of these

Answer

B

Explanation

`cot^(-1) x + cot^(-1) y + cot^(-1) z = (pi)/(2)`
or `(pi)/(2) - tan^(-1)x + (pi)/(2) - tan^(-1) y + (pi)/(2) - tan^(-1) z = (pi)/(2)`
or `tan^(-1) x + tan^(-1) y + tan^(-1) z = pi`
or `tan^(-1)x + tan^(-1) y = pi - tan^(-1) z`
or `tan(tan^(-1)x + tan^(-1) y) = tan (pi - tan^(-1) z)`
or `(x + y)/(1 -xy) = -z`
or `x + y + z = xyz`