Secant of 45 Degrees

Theorem

$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$

where $\sec$ denotes secant.

Proof

\(\ds \sec 45 \degrees\)

\(=\)

\(\ds \frac 1 {\cos 45 \degrees}\)

Secant is Reciprocal of Cosine

\(\ds \)

\(=\)

\(\ds \frac 1 {\frac {\sqrt 2} 2}\)

Cosine of $45 \degrees$

\(\ds \)

\(=\)

\(\ds \sqrt 2\)

multiplying top and bottom by $2 \sqrt 2$

$\blacksquare$

Sources

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