Tangent Half-Angle Substitution for Cosine

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Corollary to Double Angle Formula for Cosine

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

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


Proof

\(\displaystyle \cos 2 \theta\) \(=\) \(\displaystyle \cos^2 \theta - \sin^2 \theta\) Double Angle Formula for Cosine
\(\displaystyle \) \(=\) \(\displaystyle \paren {\cos^2 \theta - \sin^2 \theta} \frac {\cos^2 \theta}{\cos^2 \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {1 - \tan^2 \theta} \cos^2 \theta\) Tangent is Sine divided by Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 - \tan^2 \theta} {\sec^2 \theta}\) Secant is Reciprocal of Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 - \tan^2 \theta} {1 + \tan^2 \theta}\) Difference of Squares of Secant and Tangent

$\blacksquare$


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