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 condition given and find the minimum value of the sum of cosines.

### Step-by-Step Solution:

1. **Understanding the 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 this sum can equal zero is if each individual term is zero:
\[ \sin^2 \theta_i = 0 \quad \text{for all } i \]
This implies:
\[ \sin \theta_i = 0 \quad \text{for all } i \]

2. **Finding the Values of \(\theta_i\)**:
The sine function is zero at integer multiples of \(\pi\):
\[ \theta_i = k_i \pi \quad \text{for some integer } k_i \]
Therefore, each \(\theta_i\) can be expressed as:
\[ \theta_i = 0, \pi, 2\pi, \ldots \]

3. **Calculating \(\cos \theta_i\)**:
The cosine function takes the following values at these angles:
- \(\cos(0) = 1\)
- \(\cos(\pi) = -1\)
- \(\cos(2\pi) = 1\)
- etc.

Thus, the values of \(\cos \theta_i\) can either be \(1\) or \(-1\).

4. **Finding the Sum of Cosines**:
We need to find the minimum value of the sum:
\[ \cos \theta_1 + \cos \theta_2 + \ldots + \cos \theta_n \]
Since each \(\cos \theta_i\) can be either \(1\) or \(-1\), we can denote the number of angles where \(\cos \theta_i = -1\) as \(m\) (where \(m\) can range from \(0\) to \(n\)).

The sum can be expressed as:
\[ \text{Sum} = (n - m) \cdot 1 + m \cdot (-1) = n - 2m \]

5. **Minimizing the Sum**:
To minimize the sum \(n - 2m\), we should maximize \(m\). The maximum value of \(m\) is \(n\) (if all angles are such that \(\cos \theta_i = -1\)):
\[ \text{Minimum Sum} = n - 2n = -n \]

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