Euler's Cosine Identity/Real Domain/Proof 3

Theorem

$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$

Proof

\(\text {(1)}: \quad\)

\(\ds e^{i x}\)

\(=\)

\(\ds \cos x + i \sin x\)

Euler's Formula

\(\text {(2)}: \quad\)

\(\ds e^{-i x}\)

\(=\)

\(\ds \cos x - i \sin x\)

Euler's Formula: Corollary

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \paren {\cos x + i \sin x} + \paren {\cos x - i \sin x}\)

$(1) + (2)$

\(\ds \)

\(=\)

\(\ds 2 \cos x\)

simplifying

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \cos x\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.22)$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $22 \ \text{(a)}$