Sum of Arctangent and Arccotangent

Theorem
Let $x \in \R$ be a real number.

Then:
 * $\arctan x + \arccot x = \dfrac \pi 2$

where $\arctan$ and $\arccot$ denote arctangent and arccotangent respectively.

Proof
Let $y \in \R$ such that:
 * $\exists x \in \R: x = \map \cot {y + \dfrac \pi 2}$

Then:

Suppose $-\dfrac \pi 2 \le y \le \dfrac \pi 2$.

Then we can write:
 * $-y = \arctan x$

But then:
 * $\map \cot {y + \dfrac \pi 2} = x$

Now because $-\dfrac \pi 2 \le y \le \dfrac \pi 2$ it follows that:
 * $0 \le y + \dfrac \pi 2 \le \pi$

Hence:
 * $y + \dfrac \pi 2 = \arccot x$

That is:
 * $\dfrac \pi 2 = \arccot x + \arctan x$