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

1 Answer
jos
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#

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