Sum of Fourth Powers of Sine and Cosine

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Theorem

$\cos^4 x + \sin^4 x = 1 - 2 \cos^2 x \sin^2 x$


Proof 1

\(\ds \cos^4 x + \sin^4 x\) \(=\) \(\ds \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8 + \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8\) Power Reduction Formula for 4th Power of Cosine, Power Reduction Formula for 4th Power of Sine
\(\ds \) \(=\) \(\ds \dfrac 3 4 + \dfrac {\cos 4 x} 4\) simplification
\(\ds \) \(=\) \(\ds \dfrac 3 4 + \dfrac {8 \cos^4 x - 8 \cos^2 x + 1} 4\) Quadruple Angle Formula for Cosine
\(\ds \) \(=\) \(\ds 1 + 2 \cos^4 x - 2 \cos^2 x\) simplification
\(\ds \) \(=\) \(\ds 1 + 2 \cos^2 x \paren {1 - \sin^2 x} - 2 \cos^2 x\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 1 - 2 \sin^2 x \cos^2 x\) simplification

$\blacksquare$

Proof 2

\(\ds \cos^4 x + \sin^4 x\) \(=\) \(\ds \cos^2 x \paren {1 - \sin^2 x} + \sin^2 x \paren {1 - \cos^2 x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \cos^2 x + \sin^2 x - 2 \cos^2 x \sin^2 x\) simplification
\(\ds \) \(=\) \(\ds 1 - 2 \cos^2 x \sin^2 x\) Sum of Squares of Sine and Cosine

$\blacksquare$

Sources

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