Double Angle Formula for Cosine/Corollary 1
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Corollary to Double Angle Formula for Cosine
- $\cos 2 \theta = 2 \cos^2 \theta - 1$
where $\cos$ denotes cosine.
Proof
\(\ds \cos 2 \theta\) | \(=\) | \(\ds \cos^2 \theta - \sin^2 \theta\) | Double Angle Formula for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos^2 \theta - \paren {1 - \cos^2 \theta}\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \cos^2 \theta - 1\) |
$\blacksquare$
Also known as
This identity and Corollary $2$ are sometimes known as Carnot's Formulas, for Lazare Nicolas Marguerite Carnot.
Also see
- Square of Cosine: $\cos^2 \theta = \dfrac {1 + \cos 2 \theta} 2$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(14)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.39$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): double-angle formula
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Double-angle formulae
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Double-angle formulae