If `theta _(1), theta _(2), theta _(3) and theta_(4)` be the ecentric angles of point where a circle cuts an ellipes such that `pi lt theta _(1) + theta _(2) + theta_(3)+ theta_(4) lt 5pi,` then `cos (theta _(1) + theta _(2)+theta _(3) +theta_(4))` equals to `"______"`

Answer

To solve the problem, we need to find the value of \( \cos(\theta_1 + \theta_2 + \theta_3 + \theta_4) \) given the eccentric angles \( \theta_1, \theta_2, \theta_3, \theta_4 \) of the points where a circle intersects an ellipse, under the condition that \( \pi < \theta_1 + \theta_2 + \theta_3 + \theta_4 < 5\pi \).

### Step-by-Step Solution:

1. **Understanding Eccentric Angles**:
The angles \( \theta_1, \theta_2, \theta_3, \theta_4 \) are the angles at which the circle intersects the ellipse. Due to the symmetry of the ellipse and the circle, we can establish relationships between these angles.

2. **Symmetry Relations**:
- If \( \theta_1 \) is one angle, then the corresponding angle \( \theta_2 \) can be expressed as:
\[ \theta_2 = \pi - \theta_1 \]
- Similarly, if we let \( \theta_3 \) be another angle, then:
\[ \theta_4 = \pi - \theta_3 \]

3. **Sum of Angles**:
- Now, we can calculate the sum of all four angles:
\[ \theta_1 + \theta_2 + \theta_3 + \theta_4 = \theta_1 + (\pi - \theta_1) + \theta_3 + (\pi - \theta_3) \]
- Simplifying this gives:
\[ \theta_1 + \theta_2 + \theta_3 + \theta_4 = \pi + \pi = 2\pi \]

4. **Finding Cosine**:
- Now we need to find \( \cos(\theta_1 + \theta_2 + \theta_3 + \theta_4) \):
\[ \cos(\theta_1 + \theta_2 + \theta_3 + \theta_4) = \cos(2\pi) \]
- We know from trigonometric identities that:
\[ \cos(2\pi) = 1 \]

5. **Conclusion**:
- Therefore, the value of \( \cos(\theta_1 + \theta_2 + \theta_3 + \theta_4) \) is:
\[ \boxed{1} \]