The given expression is:

\[
\frac{(\cos x + i \sin x)^4}{(\sin x + i \cos x)^5}
\]

To simplify this expression, let's break it down step by step.

### Step 1: Use Euler's formula
Euler's formula states that:
\[
e^{ix} = \cos x + i \sin x
\]

So, we can rewrite the expression as:
\[
\frac{(e^{ix})^4}{(e^{i(\frac{\pi}{2} - x)})^5}
\]

### Step 2: Simplify the powers
\[
\frac{e^{i4x}}{e^{i5(\frac{\pi}{2} - x)}}
\]

This simplifies to:
\[
e^{i(4x - 5\left(\frac{\pi}{2} - x\right))}
\]

### Step 3: Distribute the exponent
\[
e^{i(4x - \frac{5\pi}{2} + 5x)} = e^{i(9x - \frac{5\pi}{2})}
\]

### Step 4: Use Euler's formula again
Using Euler's formula in reverse, we get:
\[
\cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)
\]

Thus, the simplified expression is:
\[
\cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)
\]

### Final Answer
\[
\boxed{\cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)}
\]

Would you like more details or have any questions?

---

1. How can Euler's formula be applied to other trigonometric expressions?
2. What is the significance of the angle \(\frac{5\pi}{2}\) in this context?
3. How would the expression change if the powers were different?
4. Can you solve a similar problem involving different trigonometric functions?
5. How does this concept relate to the complex plane?

**Tip:** Euler's formula is a powerful tool in complex analysis, connecting trigonometric functions with exponential functions.

Learn how to simplify the complex trigonometric expression (cos x + i sin x)^4 / (sin x + i cos x)^5 using Euler's formula. Understand the step-by-step process and significance of the angle \(\frac{5\pi}{2}\). Suitable for undergraduate students studying complex numbers and trigonometry.

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