Quadruple Angle Formulas/Cosine/Corollary

Corollary to Quadruple Angle Formula for Cosine

$\cos 4 \theta = 8 \sin^4 \theta - 8 \sin^2 \theta + 1$

where $\cos$ denotes cosine.

Proof

\(\ds \cos 4 \theta\)

\(=\)

\(\ds 8 \cos^4 \theta - 8 \cos^2 \theta + 1\)

Quadruple Angle Formula for Cosine

\(\ds \)

\(=\)

\(\ds 8 \paren {1 - \sin^2 \theta}^2 - 8 \paren {1 - \sin^2 \theta} + 1\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds 8 - 16 \sin^2 \theta + 8 \sin^4 \theta - 8 + 8 \sin^2 \theta + 1\)

multiplying out

\(\ds \)

\(=\)

\(\ds 8 \sin^4 \theta - 8 \sin^2 \theta + 1\)

gathering terms

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $93 \ \text{(b)}$