Sextuple Angle Formula for Tangent

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Theorem

$\tan 6 \theta = \dfrac { 6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta } { 1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta }$

where $\tan$ denotes tangent.


Proof

\(\ds \tan 6 \theta\) \(=\) \(\ds \frac {\sin 6 \theta} {\cos 6 \theta}\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac {\cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta} } {\cos 6 \theta }\) Formulation 2/Examples/Sine of Sextuple Angle
\(\ds \) \(=\) \(\ds \frac {\cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta} } {\cos^6 \theta \paren {1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta} }\) Formulation 2/Examples/Cosine of Sextuple Angle
\(\ds \) \(=\) \(\ds \frac { 6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta } { 1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta }\)

$\blacksquare$