Question

Let `f(x)=arctanx+"arccot"x`. find `f(0)+f(1)+f(sqrt 2)+ f(sqrt3)`?

Answer 1

`2 pi`

Explanation:

let `A=arctan(x), B=arccot(x)`
`tan(A)=x, cot(B)= x`
`tan(A)=cot(B)`
if `tan(A)=cot(B)=0`
`A=0, B=pi/2`
if `tan(A)=cot(B)!=0`
`tan(A)*tan(B)=1`
`1-tan(A)*tan(B)=0`
`tan(A+B)`
`=(tan(A)+tan(B)) / (1-tan(A)*tan(B))` `rightarrow infty`
`A+B=n*pi +pi/2` (n=integer number)
if `x>0`
`A+B=pi/2`(`0 < A, B < pi/2`)
if `x<0`
`A+B=pi/2`(`A < 0, pi/2 < B`)
`therefore f(x)=pi/2`
Ans=`2 pi`