If `cos 5 alpha=cos^5 alpha,` where `alpha in (0,pi/2)` then find the possible values of `(sec^2 alpha+cosec^2 alpha+cot^2 alpha).`

Answer

To solve the equation \( \cos 5\alpha = \cos^5 \alpha \) for \( \alpha \in (0, \frac{\pi}{2}) \) and find the possible values of \( \sec^2 \alpha + \csc^2 \alpha + \cot^2 \alpha \), we can follow these steps:

### Step 1: Rewrite the equation using the cosine multiple angle formula
We know that:
\[ \cos 5\alpha = 16 \cos^5 \alpha - 20 \cos^3 \alpha + 5 \cos \alpha \]
So we can rewrite the equation as:
\[ 16 \cos^5 \alpha - 20 \cos^3 \alpha + 5 \cos \alpha = \cos^5 \alpha \]

### Step 2: Rearrange the equation
Bringing all terms to one side gives us:
\[ 16 \cos^5 \alpha - \cos^5 \alpha - 20 \cos^3 \alpha + 5 \cos \alpha = 0 \]
This simplifies to:
\[ 15 \cos^5 \alpha - 20 \cos^3 \alpha + 5 \cos \alpha = 0 \]

### Step 3: Factor out the common term
Factoring out \( 5 \cos \alpha \):
\[ 5 \cos \alpha (3 \cos^4 \alpha - 4 \cos^2 \alpha + 1) = 0 \]
This gives us two cases:
1. \( 5 \cos \alpha = 0 \)
2. \( 3 \cos^4 \alpha - 4 \cos^2 \alpha + 1 = 0 \)

### Step 4: Analyze the first case
From \( 5 \cos \alpha = 0 \), we find:
\[ \cos \alpha = 0 \]
However, since \( \alpha \in (0, \frac{\pi}{2}) \), this case does not provide a valid solution.

### Step 5: Solve the second case
Let \( t = \cos^2 \alpha \). Then we rewrite the equation:
\[ 3t^2 - 4t + 1 = 0 \]
Using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \]
\[ = \frac{4 \pm \sqrt{16 - 12}}{6} = \frac{4 \pm 2}{6} \]
This results in:
\[ t = \frac{6}{6} = 1 \quad \text{or} \quad t = \frac{2}{6} = \frac{1}{3} \]

### Step 6: Find \( \cos^2 \alpha \) and \( \sin^2 \alpha \)
1. If \( \cos^2 \alpha = 1 \):
- Then \( \sin^2 \alpha = 0 \).
2. If \( \cos^2 \alpha = \frac{1}{3} \):
- Then \( \sin^2 \alpha = 1 - \frac{1}{3} = \frac{2}{3} \).

### Step 7: Calculate \( \sec^2 \alpha + \csc^2 \alpha + \cot^2 \alpha \)
Using the identities:
- \( \sec^2 \alpha = \frac{1}{\cos^2 \alpha} \)
- \( \csc^2 \alpha = \frac{1}{\sin^2 \alpha} \)
- \( \cot^2 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha} \)

**Case 1: \( \cos^2 \alpha = 1 \)**
- \( \sec^2 \alpha = 1 \)
- \( \csc^2 \alpha \) is undefined (since \( \sin^2 \alpha = 0 \)).

**Case 2: \( \cos^2 \alpha = \frac{1}{3} \)**
- \( \sec^2 \alpha = \frac{1}{\frac{1}{3}} = 3 \)
- \( \csc^2 \alpha = \frac{1}{\frac{2}{3}} = \frac{3}{2} \)
- \( \cot^2 \alpha = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \)

Now, summing these:
\[ \sec^2 \alpha + \csc^2 \alpha + \cot^2 \alpha = 3 + \frac{3}{2} + \frac{1}{2} = 3 + 2 = 5 \]

### Final Answer
The possible value of \( \sec^2 \alpha + \csc^2 \alpha + \cot^2 \alpha \) is \( \boxed{5} \).