# Cosecant of 15 Degrees

## Theorem

$\csc 15^\circ = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$

where $\csc$ denotes cosecant.

## Proof

 $\displaystyle \csc 15^\circ$ $=$ $\displaystyle \frac 1 {\sin 15^\circ}$ Cosecant is Reciprocal of Sine $\displaystyle$ $=$ $\displaystyle \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}$ Sine of 15 Degrees $\displaystyle$ $=$ $\displaystyle \frac 4 {\sqrt 6 - \sqrt 2}$ multiplying top and bottom by $4$ $\displaystyle$ $=$ $\displaystyle \frac {4 \left({\sqrt 6 + \sqrt 2}\right)} {\left({\sqrt 6 - \sqrt 2}\right) \left({\sqrt 6 + \sqrt 2}\right)}$ multiplying top and bottom by $\sqrt 6 + \sqrt 2$ $\displaystyle$ $=$ $\displaystyle \frac {4 \left({\sqrt 6 + \sqrt 2}\right)} {6 - 2}$ Difference of Two Squares $\displaystyle$ $=$ $\displaystyle \sqrt 6 + \sqrt 2$ simplifying

$\blacksquare$