If `(a costheta_1,asintheta_1),(acostheta_2,a sintheta_2)`, and `(acostheta_3a sintheta_3)` represent the vertces of an equilateral triangle inscribed in a circle. Then.

Options

A. `costheta_1+costheta_2+costheta_3=0`
B. `sintheta_1+sintheta_2+sin theta_3=0`
C. `tantheta_1+tantheta_2+tantheta_3=0`
D. `cottheta_1+cottheta_2+cottheta_3=0`

Explanation

To solve the problem, we need to show that the sum of the x-coordinates and the sum of the y-coordinates of the vertices of the equilateral triangle are both equal to zero.

### Step-by-Step Solution:

1. **Identify the Vertices**:
The vertices of the equilateral triangle are given as:
- \( A = (a \cos \theta_1, a \sin \theta_1) \)
- \( B = (a \cos \theta_2, a \sin \theta_2) \)
- \( C = (a \cos \theta_3, a \sin \theta_3) \)

2. **Find the Centroid**:
The centroid \( G \) of the triangle formed by these vertices is given by the formula:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Substituting the coordinates of the vertices:
\[ G = \left( \frac{a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3}{3}, \frac{a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3}{3} \right) \]

3. **Set the Centroid to the Origin**:
Since the triangle is inscribed in a circle and is equilateral, the centroid coincides with the circumcenter, which is at the origin (0, 0). Therefore, we set:
\[ \frac{a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3}{3} = 0 \]
\[ \frac{a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3}{3} = 0 \]

4. **Simplify the Equations**:
Multiplying both equations by 3 gives:
\[ a \cos \theta_1 + a \cos \theta_2 + a \cos \theta_3 = 0 \]
\[ a \sin \theta_1 + a \sin \theta_2 + a \sin \theta_3 = 0 \]

5. **Conclude the Results**:
Since \( a \) is a non-zero constant (the radius of the circle), we can divide both equations by \( a \):
\[ \cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 0 \]
\[ \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 0 \]

### Final Result:
Thus, we conclude that:
- The sum of the cosines of the angles is zero.
- The sum of the sines of the angles is zero.