Express the following in `a + ib` form: `((cos alpha + i sin alpha)^(4))/((sin beta + i cos beta)^(5))`

Explanation

To express the given expression \(\frac{(\cos \alpha + i \sin \alpha)^4}{(\sin \beta + i \cos \beta)^5}\) in the form \(a + ib\), we can follow these steps:

### Step 1: Rewrite using Euler's Formula
Using Euler's formula, we know that:
\[ \cos \theta + i \sin \theta = e^{i\theta} \]
Thus, we can rewrite the expression as:
\[ \frac{(\cos \alpha + i \sin \alpha)^4}{(\sin \beta + i \cos \beta)^5} = \frac{(e^{i\alpha})^4}{(e^{i(\frac{\pi}{2} - \beta)})^5} \]
This simplifies to:
\[ \frac{e^{i4\alpha}}{e^{i(5(\frac{\pi}{2} - \beta))}} = \frac{e^{i4\alpha}}{e^{i(\frac{5\pi}{2} - 5\beta)}} \]

### Step 2: Simplify the Exponent
Now, we can simplify the exponent:
\[ e^{i4\alpha} \cdot e^{-i(\frac{5\pi}{2} - 5\beta)} = e^{i(4\alpha - (\frac{5\pi}{2} - 5\beta))} \]
This can be rewritten as:
\[ e^{i(4\alpha + 5\beta - \frac{5\pi}{2})} \]

### Step 3: Convert Back to Trigonometric Form
Using Euler's formula again, we can express this in the form \(a + ib\):
\[ e^{i(4\alpha + 5\beta - \frac{5\pi}{2})} = \cos(4\alpha + 5\beta - \frac{5\pi}{2}) + i\sin(4\alpha + 5\beta - \frac{5\pi}{2}) \]

### Step 4: Simplify the Angles
We know that \(\cos(x - \frac{5\pi}{2}) = \sin x\) and \(\sin(x - \frac{5\pi}{2}) = -\cos x\). Thus:
\[ \cos(4\alpha + 5\beta - \frac{5\pi}{2}) = \sin(4\alpha + 5\beta) \]
\[ \sin(4\alpha + 5\beta - \frac{5\pi}{2}) = -\cos(4\alpha + 5\beta) \]

### Step 5: Final Expression
Putting it all together, we have:
\[ \sin(4\alpha + 5\beta) - i\cos(4\alpha + 5\beta) \]
Thus, in the form \(a + ib\):
\[ a = \sin(4\alpha + 5\beta), \quad b = -\cos(4\alpha + 5\beta) \]

### Final Result
The expression in the form \(a + ib\) is:
\[ \sin(4\alpha + 5\beta) - i\cos(4\alpha + 5\beta) \]