Power Reduction Formulas/Cosine Squared
< Power Reduction Formulas(Redirected from Square of Cosine)
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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
| \(\ds \dfrac {1 + \cos 2 x} 2\) | \(=\) | \(\ds \dfrac 1 2 \left({1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}\right)\) | Cosine Exponential Formulation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 4 \left({e^{2 i x} + 2 + e^{-2 i x} }\right)\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 4 \left({e^{2 i x} + 2 \left({e^{i x} }\right) \left({e^{-i x} }\right) + e^{-2 i x} }\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left({\dfrac {e^{i x} + e^{-i x} } 2}\right)^2\) | Square of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^2 x\) | Cosine Exponential Formulation |
$\blacksquare$
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