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.