Power Reduction Formulas/Cosine Squared
Theorem
- $\cos^2 x = \dfrac {1 + \cos 2 x} 2$
where $\cos$ denotes cosine.
Proof 1
| \(\ds 2 \cos^2 x - 1\) | \(=\) | \(\ds \cos 2 x\) | Double Angle Formula for Cosine: Corollary $1$ | |||||||||||
| \(\ds \cos^2 x\) | \(=\) | \(\ds \frac {1 + \cos 2 x} 2\) | solving for $\cos^2 x$ |
$\blacksquare$
Proof 2
| \(\ds \dfrac {1 + \cos 2 x} 2\) | \(=\) | \(\ds \dfrac 1 2 \paren {1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}\) | Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 4 \paren {e^{2 i x} + 2 + e^{-2 i x} }\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 4 \paren {e^{2 i x} + 2 \paren {e^{i x} } \paren {e^{-i x} } + e^{-2 i x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\dfrac {e^{i x} + e^{-i x} } 2}^2\) | Square of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^2 x\) | Euler's Cosine Identity |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.54$
- 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