If `sectheta+tantheta=p`. Then find the value of `tantheta`.

Explanation

To find the value of \( \tan \theta \) given that \( \sec \theta + \tan \theta = p \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Multiply and divide by \( \sec \theta - \tan \theta \):**
\[ \sec \theta + \tan \theta = p \implies \frac{(\sec \theta + \tan \theta)(\sec \theta - \tan \theta)}{\sec \theta - \tan \theta} = \frac{p(\sec \theta - \tan \theta)}{\sec \theta - \tan \theta} \]

3. **Use the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):**
\[ \sec^2 \theta - \tan^2 \theta = 1 \implies \frac{1}{\sec \theta - \tan \theta} = p \]

4. **Rearranging gives:**
\[ \sec \theta - \tan \theta = \frac{1}{p} \]

5. **Now we have two equations:**
- \( \sec \theta + \tan \theta = p \) (Equation 1)
- \( \sec \theta - \tan \theta = \frac{1}{p} \) (Equation 2)

6. **Add the two equations:**
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = p + \frac{1}{p} \]
This simplifies to:
\[ 2 \sec \theta = p + \frac{1}{p} \]
Therefore,
\[ \sec \theta = \frac{p + \frac{1}{p}}{2} \]

7. **Subtract the second equation from the first:**
\[ (\sec \theta + \tan \theta) - (\sec \theta - \tan \theta) = p - \frac{1}{p} \]
This simplifies to:
\[ 2 \tan \theta = p - \frac{1}{p} \]
Therefore,
\[ \tan \theta = \frac{p - \frac{1}{p}}{2} \]

8. **Combine the fractions:**
\[ \tan \theta = \frac{p^2 - 1}{2p} \]

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