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 values of \(\theta\) such that the eccentricity of the hyperbola given by

\[ \frac{x^2}{(1 + \sin \theta)^2} - \frac{y^2}{\cos^2 \theta} = 1 \]

is equal to \(\frac{2}{\sqrt{3}}\).

### Step 1: Identify \(a\) and \(b\)

For a hyperbola in the standard form

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \]

the eccentricity \(e\) is given by

\[ e = \sqrt{1 + \frac{b^2}{a^2}}. \]

From the given hyperbola, we can identify:

- \(a = 1 + \sin \theta\)
- \(b = \cos \theta\)

### Step 2: Write the formula for eccentricity

Substituting \(a\) and \(b\) into the eccentricity formula, we have:

\[ e = \sqrt{1 + \frac{\cos^2 \theta}{(1 + \sin \theta)^2}}. \]

### Step 3: Set the eccentricity equal to \(\frac{2}{\sqrt{3}}\)

We set the expression for eccentricity equal to the given value:

\[ \sqrt{1 + \frac{\cos^2 \theta}{(1 + \sin \theta)^2}} = \frac{2}{\sqrt{3}}. \]

### Step 4: Square both sides

Squaring both sides gives:

\[ 1 + \frac{\cos^2 \theta}{(1 + \sin \theta)^2} = \frac{4}{3}. \]

### Step 5: Simplify the equation

Subtracting 1 from both sides results in:

\[ \frac{\cos^2 \theta}{(1 + \sin \theta)^2} = \frac{4}{3} - 1 = \frac{1}{3}. \]

### Step 6: Cross-multiply

Cross-multiplying gives:

\[ 3 \cos^2 \theta = (1 + \sin \theta)^2. \]

### Step 7: Expand the right side

Expanding the right side:

\[ 3 \cos^2 \theta = 1 + 2 \sin \theta + \sin^2 \theta. \]

### Step 8: Use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\)

Substituting \(\cos^2 \theta\):

\[ 3(1 - \sin^2 \theta) = 1 + 2 \sin \theta + \sin^2 \theta. \]

### Step 9: Rearranging the equation

Rearranging gives:

\[ 3 - 3 \sin^2 \theta = 1 + 2 \sin \theta + \sin^2 \theta. \]

Combining like terms results in:

\[ 0 = 4 \sin^2 \theta + 2 \sin \theta - 2. \]

### Step 10: Simplifying the quadratic equation

Dividing the entire equation by 2:

\[ 0 = 2 \sin^2 \theta + \sin \theta - 1. \]

### Step 11: Factor the quadratic equation

Factoring gives:

\[ (2 \sin \theta + 2)(\sin \theta - 1) = 0. \]

### Step 12: Solve for \(\sin \theta\)

Setting each factor to zero gives:

1. \(2 \sin \theta + 2 = 0 \Rightarrow \sin \theta = -1\) (not in the range \(0 < \theta < \pi\))
2. \(\sin \theta - 1 = 0 \Rightarrow \sin \theta = 1 \Rightarrow \theta = \frac{\pi}{2}\)

### Step 13: Check for other possible angles

Since \(\sin \theta = \frac{1}{2}\) gives us:

\[ \theta = \frac{\pi}{6}, \, \frac{5\pi}{6} \text{ (within the range \(0 < \theta < \pi\))} \]

### Step 14: Sum of all possible values of \(\theta\)

The possible values of \(\theta\) are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\).

Calculating the sum:

\[ \frac{\pi}{6} + \frac{5\pi}{6} = \frac{6\pi}{6} = \pi. \]

### Final Answer

Thus, the sum of all possible values of \(\theta\) is:

\[ \boxed{\pi}. \]