The minimum value of `(1+1/(sin^nalpha))(1+1/(cos^nalpha))` is

Options

A. 1
B. 2
C. `(1+2^(n//2))^2`
D. 4

Answer

To find the minimum value of the expression \( (1 + \frac{1}{\sin^n \alpha})(1 + \frac{1}{\cos^n \alpha}) \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ E = \left(1 + \frac{1}{\sin^n \alpha}\right)\left(1 + \frac{1}{\cos^n \alpha}\right) \]

### Step 2: Analyze the Behavior of Sine and Cosine
We know that \( \sin \alpha \) and \( \cos \alpha \) both range between 0 and 1 for angles in the first quadrant. The expression will tend to infinity if either \( \sin \alpha \) or \( \cos \alpha \) approaches 0. Therefore, we need to find the minimum value when both \( \sin^n \alpha \) and \( \cos^n \alpha \) are positive.

### Step 3: Find the Condition for Minimum
The expression \( \sin^n \alpha \) and \( \cos^n \alpha \) will be equal when \( \sin \alpha = \cos \alpha \). This occurs at \( \alpha = \frac{\pi}{4} \).

### Step 4: Calculate Values at \( \alpha = \frac{\pi}{4} \)
At \( \alpha = \frac{\pi}{4} \):
\[ \sin \alpha = \cos \alpha = \frac{1}{\sqrt{2}} \]
Thus, we can substitute these values into the expression:
\[ E = \left(1 + \frac{1}{\left(\frac{1}{\sqrt{2}}\right)^n}\right)\left(1 + \frac{1}{\left(\frac{1}{\sqrt{2}}\right)^n}\right) \]
This simplifies to:
\[ E = \left(1 + \sqrt{2}^n\right)\left(1 + \sqrt{2}^n\right) = \left(1 + 2^{n/2}\right)^2 \]

### Step 5: Final Result
Thus, the minimum value of the expression is:
\[ \boxed{(1 + 2^{n/2})^2} \]