Tangent of 22.5 Degrees
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Theorem
- $\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$
where $\tan$ denotes tangent.
Proof 1
| \(\ds \tan 22.5 \degrees\) | \(=\) | \(\ds \tan \dfrac {45 \degrees} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 - \cos 45\degrees} {\sin 45\degrees}\) | Half Angle Formula for Tangent: Corollary $2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 - \frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}\) | Cosine of $45 \degrees$ and Sine of $45 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 2 - 1\) | multiplying top and bottom by $\sqrt 2$ |
$\blacksquare$
Proof 2
| \(\ds \tan 22.5 \degrees\) | \(=\) | \(\ds \dfrac {\sin 22.5 \degrees} {\cos 22.5 \degrees}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\dfrac 1 2 \sqrt {2 - \sqrt 2} } {\dfrac 1 2 \sqrt {2 + \sqrt 2} }\) | Sine of $22.5 \degrees$, Cosine of $22.5 \degrees$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt {2 - \sqrt 2} \sqrt {2 - \sqrt 2} } {\sqrt {2 + \sqrt 2} \sqrt {2 - \sqrt 2} }\) | multiplying top and bottom by $\sqrt {2 - \sqrt 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 - \sqrt 2} {\sqrt {2^2 - 2} }\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 2 - 1\) | simplifying |
$\blacksquare$