Cotangent of 45 Degrees

Theorem

$\cot 45 \degrees = \cot \dfrac \pi 4 = 1$

where $\cot$ denotes cotangent.

Proof

\(\ds \cot 45 \degrees\)

\(=\)

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

Cotangent is Cosine divided by Sine

\(\ds \)

\(=\)

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

Cosine of $45 \degrees$ and Sine of $45 \degrees$

\(\ds \)

\(=\)

\(\ds 1\)

dividing top and bottom by $\dfrac {\sqrt 2} 2$

$\blacksquare$

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