Double Angle Formulas/Sine/Proof 4

Theorem

$\sin 2 \theta = 2 \sin \theta \cos \theta$

Proof

\(\ds \sin 2 \theta\)

\(=\)

\(\ds \frac 1 {2 i} \paren {e^{2 i \theta} - e^{-2 i \theta} }\)

Euler's Sine Identity

\(\ds \)

\(=\)

\(\ds \frac 1 {2 i} \paren {e^{i \theta} + e^{-i \theta} } \paren {e^{i \theta} - e^{-i \theta} }\)

Difference of Two Squares

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {e^{i \theta} - e^{-i \theta} } {2 i} \cdot \frac {e^{i \theta} + e^{-i \theta} } 2}\)

\(\ds \)

\(=\)

\(\ds 2 \sin \theta \cos \theta\)

Euler's Sine Identity, Euler's Cosine Identity

$\blacksquare$