Secant of 75 Degrees

Theorem

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

where $\sec$ denotes secant.

$\ds \sec 75 \degrees$

$=$

$\ds \frac 1 {\cos 75 \degrees}$

$\ds $

$=$

$\ds \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}$

$\ds $

$=$

$\ds \frac 4 {\sqrt 6 - \sqrt 2}$

multiplying top and bottom by $4$

$\ds $

$=$

$\ds \frac {4 \paren {\sqrt 6 + \sqrt 2} } {\paren {\sqrt 6 - \sqrt 2} \paren {\sqrt 6 + \sqrt 2} }$

multiplying top and bottom by $\sqrt 6 + \sqrt 2$

$\ds $

$=$

$\ds \frac {4 \paren {\sqrt 6 + \sqrt 2} } {6 - 2}$

$\ds $

$=$

$\ds \sqrt 6 + \sqrt 2$

simplifying

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles