If e is the eccentricity of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` and `theta` is the angle between the asymptotes, then `cos.(theta)/(2)` is equal to

Options

A. `(1-e)/(e)`
B. `(2)/(e ) -e`
C. `(1)/(e )`
D. `(2)/(e )`

Explanation

To solve the problem, we need to find the value of \(\cos\left(\frac{\theta}{2}\right)\) in terms of the eccentricity \(e\) of the hyperbola given by the equation:

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

### Step-by-Step Solution:

1. **Identify the Asymptotes**:
The asymptotes of the hyperbola are given by the equations:
\[ y = \pm \frac{b}{a} x \]

2. **Determine the Angle Between the Asymptotes**:
The angle \(\theta\) between the asymptotes can be expressed as:
\[ \theta = 2 \tan^{-1}\left(\frac{b}{a}\right) \]

3. **Express \(\tan\left(\frac{\theta}{2}\right)\)**:
From the equation above, we can express \(\tan\left(\frac{\theta}{2}\right)\):
\[ \tan\left(\frac{\theta}{2}\right) = \frac{b}{a} \]

4. **Relate \(b\) and \(a\) to Eccentricity**:
The eccentricity \(e\) of the hyperbola is given by the formula:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
We can express \(\frac{b^2}{a^2}\) in terms of \(\tan\left(\frac{\theta}{2}\right)\):
\[ \frac{b^2}{a^2} = \tan^2\left(\frac{\theta}{2}\right) \]

5. **Substitute \(\tan^2\left(\frac{\theta}{2}\right)\) into the Eccentricity Formula**:
Substituting \(\tan^2\left(\frac{\theta}{2}\right)\) into the eccentricity formula gives:
\[ e^2 = 1 + \tan^2\left(\frac{\theta}{2}\right) \]

6. **Use the Identity for Secant**:
We know from trigonometric identities that:
\[ 1 + \tan^2 x = \sec^2 x \]
Thus, we can write:
\[ e^2 = \sec^2\left(\frac{\theta}{2}\right) \]
Therefore, taking the square root gives:
\[ e = \sec\left(\frac{\theta}{2}\right) \]

7. **Express \(\cos\left(\frac{\theta}{2}\right)\)**:
Since \(\sec\left(\frac{\theta}{2}\right) = \frac{1}{\cos\left(\frac{\theta}{2}\right)}\), we have:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{1}{e} \]

### Final Result:
Thus, the value of \(\cos\left(\frac{\theta}{2}\right)\) is:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{1}{e} \]