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

### Step 2: Multiply and divide by \( \sec \theta - \tan \theta \)
We multiply both sides by \( \sec \theta - \tan \theta \):
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = p(\sec \theta - \tan \theta) \]

### Step 3: Use the identity for the difference of squares
Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we get:
\[ \sec^2 \theta - \tan^2 \theta = p(\sec \theta - \tan \theta) \]

### Step 4: Apply the Pythagorean identity
Recall that \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ 1 = p(\sec \theta - \tan \theta) \]

### Step 5: Solve for \( \sec \theta - \tan \theta \)
Rearranging gives:
\[ \sec \theta - \tan \theta = \frac{1}{p} \]

### Step 6: Set up the two equations
Now we have two equations:
1. \( \sec \theta + \tan \theta = p \) (Equation 1)
2. \( \sec \theta - \tan \theta = \frac{1}{p} \) (Equation 2)

### Step 7: Subtract Equation 1 from Equation 2
Subtract Equation 1 from Equation 2:
\[ (\sec \theta - \tan \theta) - (\sec \theta + \tan \theta) = \frac{1}{p} - p \]
This simplifies to:
\[ -\tan \theta - \tan \theta = \frac{1 - p^2}{p} \]
\[ -2\tan \theta = \frac{1 - p^2}{p} \]

### Step 8: Solve for \( \tan \theta \)
Dividing both sides by -2 gives:
\[ \tan \theta = \frac{p^2 - 1}{2p} \]

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

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