Cosine Exponential Formulation/Proof 3

Theorem

$\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$

Proof

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

\(\ds \exp \paren {i z}\)

\(=\)

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

Euler's Formula

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

\(\ds \exp \paren {-i z}\)

\(=\)

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

Euler's Formula: Corollary

\(\ds \leadsto \ \ \)

\(\ds \exp \paren {i z} + \exp \paren {-i z}\)

\(=\)

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

$(1) + (2)$

\(\ds \)

\(=\)

\(\ds 2 \cos z\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds \frac {\exp \paren {i z} + \exp \paren {-i z} } 2\)

\(=\)

\(\ds \cos z\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.17)$