Deriving the world’s most Beautiful Equation

Euler’s Identity is considered to be the most beautiful equation in mathematics by many.
Euler’s Formula, the parent equation of Euler’s Identity

Taylor Series

Before we dive deep into the derivation, we shall first understand what is a Taylor Series. This will be a bit complex. So if you do not want to read this, you can go ahead and skip to the part where I talk about Exponential series, it’s expansion and the expansion of sine and cosine functions.

The Taylor expansion of a function.

Exponential Series

An exponential series is a series in the form of e^nx where n can take any real values. The most famous exponential function is e^x. But what is this e?

Definition of e as proposed by Bernoulli, and then quantified by Euler
Taylor Expansion of exponential functions

Series of Sine and Cosine

Sine and Cosine, otherwise known as sin and cos, are the two fundamental trigonometric functions.

Expansion of sin(x)
Expansion of cos(x)


Now let us get into the derivation.
We need to prove that:



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