If the eccentricity of the hyperbola `(x^(2))/((1+sin theta)^(2))-(y^(2))/(cos^(2)theta)=1` is `(2)/(sqrt3)`, then the sum of all the possible values of `theta` is (where, `theta in (0, pi)`)

Options

A. `(5pi)/(4)`
B. `(2pi)/(3)`
C. `(7pi)/(4)`
D. `pi`

Answer

To solve the problem, we need to find the sum of all possible values of \(\theta\) given the eccentricity of the hyperbola \(\frac{x^2}{(1+\sin \theta)^2} - \frac{y^2}{\cos^2 \theta} = 1\) is \(\frac{2}{\sqrt{3}}\).

### Step-by-Step Solution:

1. **Identify the parameters of the hyperbola:**
The standard form of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Here, we can identify:
\[ a = 1 + \sin \theta, \quad b = \cos \theta \]

2. **Write the formula for eccentricity:**
The eccentricity \(e\) of a hyperbola is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{\cos^2 \theta}{(1 + \sin \theta)^2}} \]

3. **Set up the equation using the given eccentricity:**
We know that:
\[ e = \frac{2}{\sqrt{3}} \]
Therefore, we can set up the equation:
\[ \sqrt{1 + \frac{\cos^2 \theta}{(1 + \sin \theta)^2}} = \frac{2}{\sqrt{3}} \]

4. **Square both sides to eliminate the square root:**
\[ 1 + \frac{\cos^2 \theta}{(1 + \sin \theta)^2} = \frac{4}{3} \]

5. **Rearrange the equation:**
\[ \frac{\cos^2 \theta}{(1 + \sin \theta)^2} = \frac{4}{3} - 1 = \frac{1}{3} \]

6. **Cross-multiply to eliminate the fraction:**
\[ 3\cos^2 \theta = (1 + \sin \theta)^2 \]

7. **Expand the right side:**
\[ 3\cos^2 \theta = 1 + 2\sin \theta + \sin^2 \theta \]

8. **Use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\):**
Substitute \(\cos^2 \theta\):
\[ 3(1 - \sin^2 \theta) = 1 + 2\sin \theta + \sin^2 \theta \]

9. **Rearrange the equation:**
\[ 3 - 3\sin^2 \theta = 1 + 2\sin \theta + \sin^2 \theta \]
\[ 0 = 4\sin^2 \theta + 2\sin \theta - 2 \]

10. **Divide the entire equation by 2:**
\[ 0 = 2\sin^2 \theta + \sin \theta - 1 \]

11. **Factor the quadratic equation:**
\[ (2\sin \theta + 2)(\sin \theta - 1) = 0 \]

12. **Solve for \(\sin \theta\):**
\[ \sin \theta = 1 \quad \text{or} \quad 2\sin \theta + 2 = 0 \quad \Rightarrow \quad \sin \theta = -1 \]
However, \(\sin \theta = -1\) is not in the interval \((0, \pi)\).

13. **Find the valid solution:**
The only valid solution is:
\[ \sin \theta = \frac{1}{2} \]
This occurs at:
\[ \theta = \frac{\pi}{6} \quad \text{and} \quad \theta = \frac{5\pi}{6} \]

14. **Sum the possible values of \(\theta\):**
\[ \text{Sum} = \frac{\pi}{6} + \frac{5\pi}{6} = \frac{6\pi}{6} = \pi \]

### Final Answer:
The sum of all possible values of \(\theta\) is \(\pi\).