Power Reduction Formulas/Tangent Squared

Theorem

$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$

where $\cos$ and $\tan$ denote cosine and tangent respectively.

Proof

\(\ds \tan^2 x\)

\(=\)

\(\ds \frac {\sin^2 x} {\cos^2 x}\)

Tangent is Sine divided by Cosine

\(\ds \)

\(=\)

\(\ds \frac {\frac {1 - \cos 2 x} 2} {\frac {\cos 2 x + 1} 2}\)

Square of Sine and Square of Cosine

\(\ds \)

\(=\)

\(\ds \frac {1 - \cos 2 x} {1 + \cos 2 x}\)

multiplying top and bottom by $2$

$\blacksquare$