Tangent Half-Angle Substitution for Cosine
(Redirected from Double Angle Formula for Cosine/Corollary 3)
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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
\(\ds \cos 2 \theta\) | \(=\) | \(\ds \cos^2 \theta - \sin^2 \theta\) | Double Angle Formula for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\cos^2 \theta - \sin^2 \theta} \frac {\cos^2 \theta}{\cos^2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - \tan^2 \theta} \cos^2 \theta\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 - \tan^2 \theta} {\sec^2 \theta}\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 - \tan^2 \theta} {1 + \tan^2 \theta}\) | Difference of Squares of Secant and Tangent |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(27)$