# Power Reduction Formulas/Sine Cubed

## Theorem

$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$

where $\sin$ denotes sine.

## Proof 1

 $\displaystyle \sin 3 x$ $=$ $\displaystyle 3 \sin x - 4 \sin^3 x$ Triple Angle Formula for Sine $\displaystyle \implies \ \$ $\displaystyle 4 \sin^3 x$ $=$ $\displaystyle 3 \sin x - \sin 3 x$ rearranging $\displaystyle \implies \ \$ $\displaystyle \sin^3 x$ $=$ $\displaystyle \frac {3 \sin x - \sin 3 x} 4$ dividing both sides by $4$

$\blacksquare$

## Proof 2

 $\displaystyle \sin^3 x$ $=$ $\displaystyle \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^3$ Sine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac {\paren {e^{i x} - e^{-i x} }^3} {8 i^3}$ rearranging $\displaystyle$ $=$ $\displaystyle -\frac 1 {8 i} \paren {\paren {e^{i x} }^3 - 3 \paren {e^{i x} }^2 \paren {e^{-i x} } + 3 \paren {e^{i x} } \paren {e^{-i x} }^2 - \paren {e^{-i x} }^3}$ multiplying out $\displaystyle$ $=$ $\displaystyle -\frac 1 {8 i} \paren {e^{3 i x} - 3 e^{i x} + 3 e^{-i x} - e^{-3 i x} }$ simplifying $\displaystyle$ $=$ $\displaystyle \frac 3 4 \paren {\frac {e^{i x} - e^{-i x} } {2 i} } - \frac 1 4 \paren {\frac {e^{3 i x} - e^{-3 i x} } {2 i} }$ gathering terms $\displaystyle$ $=$ $\displaystyle \frac {3 \sin x - \sin 3 x} 4$ Sine Exponential Formulation

$\blacksquare$