You're correct. Let's revisit the problem to obtain the correct simplified expression.

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

### Step 1: Use Euler's formula
Euler's formula states that:
\[
\cos x + i \sin x = e^{ix}
\]
\[
\sin x + i \cos x = e^{i\left(\frac{\pi}{2} - x\right)}
\]

### Step 2: Express the given expression using Euler's formula
The expression can be rewritten as:
\[
\frac{(e^{ix})^4}{(e^{i\left(\frac{\pi}{2} - x\right)})^5}
\]

### Step 3: Simplify the powers
This simplifies to:
\[
\frac{e^{i4x}}{e^{i\left(\frac{5\pi}{2} - 5x\right)}} = e^{i\left(4x - \left(\frac{5\pi}{2} - 5x\right)\right)}
\]
Simplifying the exponent:
\[
e^{i(4x - \frac{5\pi}{2} + 5x)} = e^{i(9x - \frac{5\pi}{2})}
\]

### Step 4: Use the identity for phase shifts
We know:
\[
e^{i(9x - \frac{5\pi}{2})} = \cos\left(9x - \frac{5\pi}{2}\right) + i\sin\left(9x - \frac{5\pi}{2}\right)
\]

However, \(\frac{5\pi}{2} = 2\pi + \frac{\pi}{2}\), so:
\[
e^{i(9x - \frac{5\pi}{2})} = \cos\left(9x - \frac{\pi}{2}\right) + i\sin\left(9x - \frac{\pi}{2}\right)
\]

Using the identity \(\cos\left(\theta - \frac{\pi}{2}\right) = \sin\theta\) and \(\sin\left(\theta - \frac{\pi}{2}\right) = -\cos\theta\), we get:
\[
\cos\left(9x - \frac{\pi}{2}\right) + i\sin\left(9x - \frac{\pi}{2}\right) = \sin 9x - i \cos 9x
\]

### Final Answer:
\[
\boxed{\sin 9x - i \cos 9x}
\]

Would you like more details or have any questions?

---

1. How can phase shifts affect trigonometric expressions?
2. What is the significance of using \(\frac{5\pi}{2}\) in trigonometric functions?
3. How does the identity for \(\cos\left(\theta - \frac{\pi}{2}\right)\) work?
4. Can this method be applied to other complex trigonometric expressions?
5. How does this relate to the concept of rotations in the complex plane?

**Tip:** When simplifying complex expressions, always check if trigonometric identities apply after simplifying the exponent to ensure the correct final form.

Learn how to simplify the complex trigonometric expression sin 9x - i cos 9x step-by-step using Euler's formula and trigonometric identities. This problem involves complex numbers and advanced trigonometric concepts.

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