Let's solve the expression:

\[
\frac{(\cos x - i \sin x)^{10}}{(\cos x + i \sin x)^{12}}
\]

First, recall the Euler's formula:

\[
\cos x + i \sin x = e^{ix} \quad \text{and} \quad \cos x - i \sin x = e^{-ix}
\]

Substituting these into the expression:

\[
\frac{(e^{-ix})^{10}}{(e^{ix})^{12}} = \frac{e^{-10ix}}{e^{12ix}}
\]

We can simplify this further by combining the exponents:

\[
e^{-10ix - 12ix} = e^{-22ix}
\]

Thus, the expression simplifies to:

\[
e^{-22ix}
\]

Finally, using Euler's formula in reverse:

\[
e^{-22ix} = \cos(22x) - i\sin(22x)
\]

So, the simplified form of the given expression is:

\[
\cos(22x) - i\sin(22x)
\]

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How does Euler's formula relate trigonometric functions to exponential functions?
2. What happens if you raise \(\cos x + i \sin x\) to a different power, say 15?
3. Can you express \(\sin(22x)\) and \(\cos(22x)\) in terms of simpler trigonometric functions?
4. What are the properties of complex conjugates and how do they apply here?
5. How would you evaluate \(\frac{(\cos x - i \sin x)^5}{(\cos x + i \sin x)^3}\)?

**Tip:** Euler's formula is a powerful tool in complex number analysis, particularly useful for converting between exponential and trigonometric forms.

Learn how to simplify the complex expression (cos x - i sin x)^10 / (cos x + i sin x)^12 using Euler's formula. Understand the connection between trigonometric functions and exponential functions in complex analysis. Suitable for advanced undergraduate students and anyone interested in complex numbers and Euler's formula.

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