# How do you evaluate sec(sec^-1((2sqrt3)/3)) without a calculator?

Sep 22, 2016

$2 \frac{\sqrt{3}}{3}$

#### Explanation:

Use $f {f}^{- 1} \left(c\right) = c$, for any function f that is locally bijective (single-

valued either way), at x = c.

For that matter,

$\sin \sin \left(- 1\right) \left(c\right)$

=cos cos^-1)(c)

$= \tan {\tan}^{- 1} \left(c\right)$

$= \csc {\csc}^{- 1} \left(c\right)$

$= \sec {\sec}^{- 1} \left(c\right)$

$= \cot {\cot}^{- 1} \left(c\right)$

$= \ln {e}^{c}$

$= {\log}_{b} {b}^{c}$

$= c$.

Here,

$\sec {\sec}^{- 1} \left(2 \frac{\sqrt{3}}{3}\right) = 2 \frac{\sqrt{3}}{3}$.