Secant of 75 Degrees

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Theorem

$\sec 75^\circ = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$

where $\sec$ denotes secant.


Proof

\(\displaystyle \sec 75^\circ\) \(=\) \(\displaystyle \frac 1 {\cos 75^\circ}\) Secant is Reciprocal of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}\) Cosine of 75 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

$\blacksquare$


Sources