# Power Reduction Formulas/Sine to 4th

## Theorem

$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$

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

## Proof

 $\ds \sin^4 x$ $=$ $\ds \paren {\sin^2 x}^2$ $\ds$ $=$ $\ds \paren {\frac {1 - \cos 2 x} 2}^2$ Square of Sine $\ds$ $=$ $\ds \frac {1 - 2 \cos 2 x + \cos^2 2 x} 4$ multiplying out $\ds$ $=$ $\ds \frac {1 - 2 \cos 2 x + \frac {1 + \cos 4 x} 2} 4$ Square of Cosine $\ds$ $=$ $\ds \frac {2 - 4 \cos 2 x + 1 + \cos 4 x} 8$ multiplying top and bottom by $2$ $\ds$ $=$ $\ds \frac {3 - 4 \cos 2 x + \cos 4 x} 8$ rearrangement

$\blacksquare$