Power Reduction Formulas/Cosine Squared

Theorem

$\cos^2 x = \dfrac {1 + \cos 2 x} 2$

where $\cos$ denotes cosine.

Proof 1

 $\displaystyle 2 \cos^2 x - 1$ $=$ $\displaystyle \cos 2 x$ Double Angle Formula for Cosine: Corollary 1 $\displaystyle \cos^2 x$ $=$ $\displaystyle \frac {1 + \cos 2 x} 2$ solving for $\cos^2 x$

$\blacksquare$

Proof 2

 $\displaystyle \dfrac {1 + \cos 2 x} 2$ $=$ $\displaystyle \dfrac 1 2 \left({1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}\right)$ Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \dfrac 1 4 \left({e^{2 i x} + 2 + e^{-2 i x} }\right)$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac 1 4 \left({e^{2 i x} + 2 \left({e^{i x} }\right) \left({e^{-i x} }\right) + e^{-2 i x} }\right)$ $\displaystyle$ $=$ $\displaystyle \left({\dfrac {e^{i x} + e^{-i x} } 2}\right)^2$ Square of Sum $\displaystyle$ $=$ $\displaystyle \cos^2 x$ Cosine Exponential Formulation

$\blacksquare$