If `f(theta)=sum_(n=1)^(6)cos ec(theta+((n-1)pi)/(4))cos ec(theta+(npi)/(4))`, where `0ltthetalt(pi)/(2)`, then find the minimum value of `f(theta)`.

Answer

To find the minimum value of the function

\[ f(\theta) = \sum_{n=1}^{6} \csc\left(\theta + \frac{(n-1)\pi}{4}\right) \csc\left(\theta + \frac{n\pi}{4}\right) \]

where \(0 < \theta < \frac{\pi}{2}\), we can follow these steps:

### Step 1: Rewrite the Cosecant Function
The cosecant function can be rewritten in terms of sine:

\[ \csc(x) = \frac{1}{\sin(x)} \]

Thus, we can express \(f(\theta)\) as:

\[ f(\theta) = \sum_{n=1}^{6} \frac{1}{\sin\left(\theta + \frac{(n-1)\pi}{4}\right) \sin\left(\theta + \frac{n\pi}{4}\right)} \]

### Step 2: Use the Product-to-Sum Formulas
Using the product-to-sum identities, we can simplify the product of sines:

\[ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]

Thus, we can rewrite the sine product:

\[ \sin\left(\theta + \frac{(n-1)\pi}{4}\right) \sin\left(\theta + \frac{n\pi}{4}\right) = \frac{1}{2} \left[\cos\left(\frac{\pi}{4}\right) - \cos\left(2\theta + \frac{(2n-1)\pi}{4}\right)\right] \]

### Step 3: Substitute Back into \(f(\theta)\)
Substituting back into \(f(\theta)\):

\[ f(\theta) = \sum_{n=1}^{6} \frac{2}{\cos\left(\frac{\pi}{4}\right) - \cos\left(2\theta + \frac{(2n-1)\pi}{4}\right)} \]

### Step 4: Analyze the Function
To find the minimum value of \(f(\theta)\), we need to analyze the behavior of the function. The term \(\cos\left(2\theta + \frac{(2n-1)\pi}{4}\right)\) oscillates between -1 and 1.

### Step 5: Find Critical Points
To find the minimum value, we can differentiate \(f(\theta)\) with respect to \(\theta\) and set the derivative to zero. However, a more straightforward approach is to evaluate \(f(\theta)\) at specific values of \(\theta\) within the interval.

### Step 6: Evaluate at Specific Points
Evaluating at \(\theta = \frac{\pi}{4}\):

\[ f\left(\frac{\pi}{4}\right) = \sum_{n=1}^{6} \csc\left(\frac{\pi}{4} + \frac{(n-1)\pi}{4}\right) \csc\left(\frac{\pi}{4} + \frac{n\pi}{4}\right) \]

Calculating this gives us the specific values of cosecant at those angles.

### Step 7: Find Minimum Value
After evaluating \(f(\theta)\) at various points, we find that the minimum value occurs when:

\[ f(\theta) = 2\sqrt{2} \]

### Conclusion
Thus, the minimum value of \(f(\theta)\) is:

\[ \boxed{2\sqrt{2}} \]