Could someone explain me this "ownership" of the arctangent

Question

someone could explain to me this:

$$\int { \arctan { \left( \frac { 1 }{ { u }^{ 2 } } \right) } } \,du=\int { \frac { \pi }{ 2 } } -\arctan { \left( { u }^{ 2 } \right) } \, du$$

Answer 1

If you draw a right triangle with sides $a,b,c$ angle $A$ opposite leg $a$, you have $\arctan \frac ab=A, \arctan \frac ba=\frac \pi 2-A$. The general relation then is $\arctan \frac 1x=\frac \pi 2 -\arctan x$. The integral signs don't matter.

Answer 2

$$\boxed{\tan^{-1} \dfrac 1 x = \cot^{-1}x = \dfrac {\pi} 2 - \tan^{-1} x}$$

Proof of first equality: $y = \tan^{-1} \dfrac 1 x \Rightarrow \tan y = \dfrac 1 x \Rightarrow \cot y = x \Rightarrow y = \cot^{-1} x$.

Proof of second equality: $\cot \left(\dfrac {\pi} 2 - \theta\right) = \tan \theta \Rightarrow \cot \left(\dfrac {\pi} 2 - \tan^{-1} x \right) = \tan \tan^{-1}x = x \Rightarrow \dfrac {\pi} 2 - \tan^{-1} x = \cot^{-1} x$.