Triple Angle Formulas/Tangent/Corollary 1

Corollary to Triple Angle Formula for Tangent

$\tan 3 \theta = \tan \theta \dfrac {4 \cos^2 \theta - 1} {4 \cos^2 \theta - 3}$

$\ds \tan 3 \theta$

$=$

$\ds \frac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$

$\ds $

$=$

$\ds \tan \theta \frac {3 - \tan^2 \theta} {1 - 3 \tan^2 \theta}$

$\ds $

$=$

$\ds \tan \theta \frac {3 - \paren {\sec^2 \theta -1 } } {1 - 3 \paren {\sec^2 \theta - 1} }$

$\ds $

$=$

$\ds \tan \theta \frac {4 - \sec^2 \theta} {4 - 3\sec^2 \theta}$

$\ds $

$=$

$\ds \tan \theta \dfrac {4 \cos^2 \theta - 1} {4 \cos^2 \theta - 3}$

$\blacksquare$