If `sec theta + tan theta= p`, then `costheta` is

Options

A. `(p^2 + 1)/(p^2 -1)`
B. `(p^2 -1)/((p^2 + 1)^2)`
C. `(2p)/(p^2 + 1)`
D. `(4p^2)/((p^2 + 1)^2)`

Answer

To solve the problem, we need to find the value of \( \cos \theta \) given that \( \sec \theta + \tan \theta = p \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sec \theta + \tan \theta = p \]

2. **Use the identity**:
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

3. **Substitute \( \sec \theta + \tan \theta \)**:
From the given equation, we can substitute:
\[ (\sec \theta - \tan \theta)(p) = 1 \]
Therefore:
\[ \sec \theta - \tan \theta = \frac{1}{p} \]

4. **Now we have two equations**:
We have:
\[ \sec \theta + \tan \theta = p \quad \text{(1)} \]
\[ \sec \theta - \tan \theta = \frac{1}{p} \quad \text{(2)} \]

5. **Add equations (1) and (2)**:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = p + \frac{1}{p} \]
This simplifies to:
\[ 2\sec \theta = p + \frac{1}{p} \]
Thus:
\[ \sec \theta = \frac{p + \frac{1}{p}}{2} \]

6. **Find \( \cos \theta \)**:
Since \( \cos \theta = \frac{1}{\sec \theta} \), we can substitute:
\[ \cos \theta = \frac{2}{p + \frac{1}{p}} \]
To simplify this, multiply numerator and denominator by \( p \):
\[ \cos \theta = \frac{2p}{p^2 + 1} \]

### Final Result:
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{2p}{p^2 + 1} \]