Question

Let $f(x) = \arctan x + \mathop{\text{arccot}} x$. Find $f(0) + f(1) + f(\sqrt{2}) + f(\sqrt{3})$.

Answer 1

By direct computation, arctan(0) = 0, arccot(0) = pi/2, arctan(1) = arccot(1) = pi/4.

Let a = arctan(sqrt(2)). Then cot(3*pi/2 - a) = cot(pi/2 - a) = tan(a) = sqrt(2), so arccot(sqrt(2)) = 3*pi/2 - a. Therefore, arctan(sqrt(2)) + arccot(sqrt(2)) = 3*pi/2.

Similarly, arctan(sqrt(3)) + arccot(sqrt(3)) = 3*pi/2.

So the answer is 0 + pi/2 + pi/4 + pi/4 + 3*pi/2 + 3*pi/2 = 4*pi.