If `sin^2theta_1+sin^2theta_2+...+sin^2theta_n=0`, then find the minimum value of `costheta_1+costheta_2+...+costheta_n`.

Explanation

To solve the problem, we need to analyze the given condition and find the minimum value of the sum of cosines.

### Step 1: Analyze the given condition
We are given that:
\[ \sin^2 \theta_1 + \sin^2 \theta_2 + \ldots + \sin^2 \theta_n = 0 \]
Since \(\sin^2 \theta_i \geq 0\) for all \(i\), the only way for this sum to equal zero is if each individual term is zero. Therefore, we have:
\[ \sin^2 \theta_i = 0 \quad \text{for all } i \]
This implies:
\[ \sin \theta_i = 0 \quad \text{for all } i \]
Thus, \(\theta_i\) must be an integer multiple of \(\pi\):
\[ \theta_i = k_i \pi \quad \text{for some integer } k_i \]

### Step 2: Find the values of \(\cos \theta_i\)
Using the fact that \(\theta_i = k_i \pi\), we can find the cosine values:
\[ \cos \theta_i = \cos(k_i \pi) \]
The cosine of integer multiples of \(\pi\) is:
\[ \cos(k_i \pi) = (-1)^{k_i} \]
This means that \(\cos \theta_i\) can either be \(1\) or \(-1\) depending on whether \(k_i\) is even or odd.

### Step 3: Calculate the sum of cosines
We need to find the minimum value of:
\[ \cos \theta_1 + \cos \theta_2 + \ldots + \cos \theta_n \]
Given that each \(\cos \theta_i\) can be either \(1\) or \(-1\), the minimum value occurs when all \(\cos \theta_i\) are \(-1\):
\[ \cos \theta_1 = \cos \theta_2 = \ldots = \cos \theta_n = -1 \]
Thus, the sum becomes:
\[ \cos \theta_1 + \cos \theta_2 + \ldots + \cos \theta_n = -1 - 1 - \ldots - 1 \quad (n \text{ times}) = -n \]

### Conclusion
The minimum value of \( \cos \theta_1 + \cos \theta_2 + \ldots + \cos \theta_n \) is:
\[ \boxed{-n} \]