Power Reduction Formulas/Cosine Squared/Proof 2

Theorem

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

Proof

\(\ds \dfrac {1 + \cos 2 x} 2\)

\(=\)

\(\ds \dfrac 1 2 \paren {1 + \dfrac {e^{2 i x} + e^{-2 i x} } 2}\)

Euler's Cosine Identity

\(\ds \)

\(=\)

\(\ds \dfrac 1 4 \paren {e^{2 i x} + 2 + e^{-2 i x} }\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac 1 4 \paren {e^{2 i x} + 2 \paren {e^{i x} } \paren {e^{-i x} } + e^{-2 i x} }\)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac {e^{i x} + e^{-i x} } 2}^2\)

Square of Sum

\(\ds \)

\(=\)

\(\ds \cos^2 x\)

Euler's Cosine Identity

$\blacksquare$