Power Reduction Formulas/Sine Squared

Theorem

$\sin^2 x = \dfrac {1 - \cos 2 x} 2$

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

Proof

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

\(=\)

\(\ds \cos 2 x\)

Double Angle Formula for Cosine: Corollary $2$

\(\ds \leadsto \ \ \)

\(\ds \sin^2 x\)

\(=\)

\(\ds \frac {\cos 2 x - 1} {-2}\)

solving for $\sin^2x$

\(\ds \)

\(=\)

\(\ds \frac {1 - \cos 2 x} 2\)

multiplying top and bottom by $-1$ and rearranging terms

$\blacksquare$

Historical Note

The Square of Sine formula was discovered and documented by Varahamihira in the $6$th century CE.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.53$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Double-angle formulae
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Double-angle formulae