# Power Reduction Formulas/Sine Squared

< Power Reduction Formulas(Redirected from Square of Sine)

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## Contents

## Theorem

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

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

## Proof

\(\displaystyle 1 - 2 \sin^2 x\) | \(=\) | \(\displaystyle \cos 2 x\) | Double Angle Formula for Cosine: Corollary 2 | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sin^2 x\) | \(=\) | \(\displaystyle \frac {\cos 2 x - 1} {-2}\) | solving for $\sin^2x$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {1 - \cos 2 x} 2\) | 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: $5.53$