Double Angle Formulas/Cosine/Corollary 1

Corollary to Double Angle Formula for Cosine

$\cos 2 \theta = 2 \cos^2 \theta - 1$

where $\cos$ denotes cosine.

Proof

\(\ds \cos 2 \theta\)

\(=\)

\(\ds \cos^2 \theta - \sin^2 \theta\)

Double Angle Formula for Cosine

\(\ds \)

\(=\)

\(\ds \cos^2 \theta - \paren {1 - \cos^2 \theta}\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds 2 \cos^2 \theta - 1\)

$\blacksquare$

Also known as

This identity and Corollary $2$ are sometimes known as Carnot's Formulas, for Lazare Nicolas Marguerite Carnot.

Also see

• Square of Cosine: $\cos^2 \theta = \dfrac {1 + \cos 2 \theta} 2$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(14)$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.39$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: double-angle formula