Sextuple Angle Formulas/Tangent

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

\(\ds \)

\(=\)

\(\ds \frac { 6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta } { 1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta }\)

$\blacksquare$