Power Reduction Formulas/Sine Squared

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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