Sum of Arcsecant and Arccosecant

Theorem
Let $x \in \R$ be a real number such that $\size x \ge 1$.

Then:
 * $\arcsec x + \arccsc x = \dfrac \pi 2$

where $\arcsec$ and $\arccsc$ denote arcsecant and arccosecant respectively.

Proof
Let $y \in \R$ such that:
 * $\exists x \in \R: \size x \ge 1$ and $x = \map \csc {y + \dfrac \pi 2}$

Then:

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

Then we can write $-y = \arccsc x$.

But then $\map \csc {y + \dfrac \pi 2} = x$.

Now since $-\dfrac \pi 2 \le y \le \dfrac \pi 2$ it follows that $0 \le y + \dfrac \pi 2 \le \pi$.

Hence $y + \dfrac \pi 2 = \arcsec x$.

That is, $\dfrac \pi 2 = \arcsec x + \arccsc x$.