Secant of 75 Degrees

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Theorem

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

where $\sec$ denotes secant.


Proof

\(\displaystyle \sec 75 \degrees\) \(=\) \(\displaystyle \frac 1 {\cos 75 \degrees}\) 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 $4$
\(\displaystyle \) \(=\) \(\displaystyle \frac {4 \paren {\sqrt 6 + \sqrt 2} } {\paren {\sqrt 6 - \sqrt 2} \paren {\sqrt 6 + \sqrt 2} }\) multiplying top and bottom by $\sqrt 6 + \sqrt 2$
\(\displaystyle \) \(=\) \(\displaystyle \frac {4 \paren {\sqrt 6 + \sqrt 2} } {6 - 2}\) Difference of Two Squares
\(\displaystyle \) \(=\) \(\displaystyle \sqrt 6 + \sqrt 2\) simplifying

$\blacksquare$


Sources