Euler's formula is given by:

\[
e^{ix} = \cos(x) + i\sin(x)
\]

Now, applying this to \(1 - e^{ix}\), we get:

\[
1 - e^{ix} = 1 - (\cos(x) + i\sin(x))
\]
\[
1 - e^{ix} = (1 - \cos(x)) - i\sin(x)
\]

This expression separates into real and imaginary parts, where the real part is \(1 - \cos(x)\) and the imaginary part is \(-\sin(x)\).

Would you like to explore this in more detail, or perhaps discuss applications of Euler's formula?

Here are 5 related questions to consider:
1. How is Euler's formula used in complex number calculations?
2. Can Euler's formula be extended to other trigonometric identities?
3. What is the geometric interpretation of \(1 - e^{ix}\)?
4. How does the magnitude of \(1 - e^{ix}\) change with different values of \(x\)?
5. How is Euler's formula applied in solving differential equations?

**Tip**: Euler's formula is a powerful tool for converting between exponential and trigonometric forms, making it highly useful in physics and engineering!

Explore the application of Euler's formula to solve the expression 1 - e^ix. Learn how to separate the real and imaginary parts of the equation, and understand its implications in complex number theory and trigonometry. Suitable for undergraduate mathematics students.

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