Arccosine in terms of Arctangent

Theorem

 * $\displaystyle \arccos x = 2 \map \arctan {\sqrt {\frac {1 - x} {1 + x} } }$

where $x$ is a real number with $-1 < x \le 1$

Proof
Let:


 * $\theta = \arccos x$

Then:


 * $x = \cos \theta$

and:


 * $0 \le \theta < \pi$

by the definition of arccosine.

Then: