Composition of Trigonometric Functions and their Inverses

Theorem: Functions of Arcfunctions

The compositions of real inverse trigonometric functions inside real trigonometric functions have the following properties:

$$\begin{array}{rl}& \cos (\arctan (x))=\frac{1}{\sqrt{1+x^2}}\\ & \cos (\arcsin (x))=\frac{1}{\sqrt{1-x^2}}\\ & \cos \left(\arctan \left(\frac{1}{x}\right)\right)=\frac{|x|}{\sqrt{1+x^2}}\end{array}$$

PROOF

TODO