Double Angle Formulas/Cosine/Corollary 2

Corollary to Double Angle Formula for Cosine

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

where $\cos$ and $\sin$ denote cosine and sine respectively.

Proof

\(\ds \cos 2 \theta\)

\(=\)

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

Double Angle Formula for Cosine

\(\ds \)

\(=\)

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

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

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

$\blacksquare$

Also known as

This identity and Corollary $1$ are sometimes known as Carnot's Formulas, for Lazare Nicolas Marguerite Carnot.

Also see

• Square of Sine: $\sin^2 \theta = \dfrac {1 - \cos 2 \theta} 2$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(15)$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.39$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Double-angle formulae