# Double Angle Formulas/Cosine/Proof 3

$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$
 $\displaystyle \cos^2 \theta - \sin^2\theta$ $=$ $\displaystyle \left({\frac 1 2 \left({e^{i \theta} + e^{-i \theta} }\right)}\right)^2 - \left({\frac 1 {2 i} \left({e^{i \theta} - e^{-i \theta} }\right)}\right)^2$ Cosine Exponential Formulation, Sine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac 1 4 \left({e^{i \theta} + e^{-i \theta} }\right)^2 + \frac 1 4 \left({e^{i \theta} - e^{-i \theta} }\right)^2$ $i$ is the imaginary unit $\displaystyle$ $=$ $\displaystyle \frac 1 4 \left({e^{2 i \theta} + 2 + e^{-2 i \theta} + e^{2 i \theta} - 2 + e^{-2 i \theta} }\right)$ Square of Sum, Square of Difference $\displaystyle$ $=$ $\displaystyle \frac 1 2 \left({e^{2 i \theta} + e^{-2 i \theta} }\right)$ Simplifying $\displaystyle$ $=$ $\displaystyle \cos 2 \theta$ Cosine Exponential Formulation
$\blacksquare$