Simple Complex numbers

$\large \dfrac2{1 - e^{i \pi /5}} = \dfrac{e^{i \cdot 2\pi /5}}{\sin \frac\pi{10}}$

Prove the equation above.

#Geometry

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$e^{\frac{i\pi}{5}}=\cos \frac{\pi}{5}+i\sin \frac{\pi}{5}\\ 1-\cos \frac{\pi}{5}=2\sin^2\frac{\pi}{10}\\ \sin \frac{\pi}{5}=2\sin \frac{\pi}{10}\cos \frac{\pi}{10}$ such that given expression simplifies to:- $\dfrac{1}{\sin \frac{\pi}{10}(\cos (\frac{-2\pi}{5})+i\sin (\frac{-2\pi}{5}))}$ $\large =\dfrac{e^{i \cdot 2\pi /5}}{\sin \frac\pi{10}}$

Rishabh Jain - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$