Sum of Squares of Sine and Cosine/Proof 5

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Theorem

$\cos^2 x + \sin^2 x = 1$


Proof

\(\displaystyle \cos^2 x + \sin^2 x\) \(=\) \(\displaystyle \paren {\frac {e^{i x} + e^{-i x} } 2}^2 + \sin^2 x\) Cosine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \paren {\frac {e^{i x} + e^{-i x} } 2}^2 + \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^2\) Sine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i x} }^2 + 2 e^{-i x} e^{i x} + \paren {e^{-i x} }^2} 4 + \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^2\) Square of Sum
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i x} }^2 + 2 e^{-i x} e^{i x} + \paren {e^{-i x} }^2} 4 + \frac {\paren {e^{i x} }^2 - e^{-i x} e^{i x} + \paren {e^{-i x} }^2} {-4}\) Square of Difference and $i^2 = -1$
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{2 i x} + 2 + e^{-2 i x} } 4 + \frac {e^{2 i x} - 2 + e^{-2 i x} } {-4}\) Exponential of Sum
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{2 i x} + 2 + e^{-2 i x} - e^{2 i x} + 2 - e^{-2 i x} } 4\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 4 4\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$


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