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

1 Answer
Jun 18, 2018

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