Power Reduction Formulas/Cosine Squared

From ProofWiki
Jump to navigation Jump to search

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

$\blacksquare$


Sources