Power Reduction Formulas/Cosine to 4th
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Theorem
- $\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$
where $\cos$ denotes cosine.
Proof 1
\(\ds \cos^4 x\) | \(=\) | \(\ds \paren {\cos^2 x}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {1 + \cos 2 x} 2}^2\) | Square of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 + 2 \cos 2 x + \cos^2 2 x} 4\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 + 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4\) | Square of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 + 4 \cos 2 x + 1 + \cos 4 x} 8\) | multiplying top and bottom by $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cos 2 x + \cos 4 x} 8\) | rearrangement |
$\blacksquare$
Proof 2
\(\ds \cos ^4 x\) | \(=\) | \(\ds \paren {\frac {e^{i x} + e^{-i x} } 2}^4\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i x} + e^{-i x} }^4} {16}\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i x} }^4 + 4 \paren {e^{i x} }^3 \paren {e^{-i x} } + 6 \paren {e^{i x} }^2 \paren {e^{-i x} }^2 + 4 \paren {e^{i x} } \paren {e^{-i x} }^3 + \paren {e^{-i x} }^4} {16}\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{4 i x} + 4 e^{2 i x} + 6 + 4 e^{-2 i x} + e^{-4 i x} } {16}\) | multiplying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \paren {\dfrac {e^{2 i x} + e^{-2 i x} } 2} + \paren {\dfrac {e^{4 i x} + e^{-4 i x} } 2} } 8\) | gathering terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cos 2 x + \cos 4 x} 8\) | Euler's Cosine Identity |
$\blacksquare$
Also defined as
This result can often be seen as:
- $\cos^4 x = \dfrac 3 8 + \dfrac {\cos 2 x} 2 + \dfrac {\cos 4 x} 8$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.58$