To prove Euler's formula, $e^{ix} = \cos x + i \sin x$, we'll use the Taylor series expansions for the exponential function, cosine, and sine.

Step 1: Taylor Series Expansions

The Taylor series expansion for $e^{ix}$ is:

$e^{ix} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \cdots$

Simplifying each term:

$e^{ix} = 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

We can separate this into the real and imaginary parts:

$e^{ix} = \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \right)$

Step 2: Compare with Cosine and Sine Expansions

The Taylor series expansion for $\cos x$ is:

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

The Taylor series expansion for $\sin x$ is:

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

Step 3: Combine the Results

Notice that the real part of the expansion of $e^{ix}$ matches the Taylor series for $\cos x$, and the imaginary part matches the Taylor series for $\sin x$. Therefore:

$e^{ix} = \cos x + i \sin x$

This completes the proof of Euler's formula.

Would you like further details on any part of this proof? Here are some related questions:

1. How do Taylor series expansions work?
2. Can you prove Euler's formula using a different method?
3. How does Euler's formula relate to the unit circle?
4. What are the applications of Euler's formula in physics?
5. How can we use Euler's formula to derive trigonometric identities?

Tip: Euler's formula is a cornerstone in complex analysis, linking trigonometry and exponential functions in a profound way.