If `sec theta + tan theta =p,` then `tan theta ` is equal to

Options

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

Answer

To solve the equation \( \sec \theta + \tan \theta = p \) for \( \tan \theta \), we can follow these steps:

### Step 1: Rewrite the equation
Given:
\[ \sec \theta + \tan \theta = p \]

### Step 2: Use the identity for secant
We know that:
\[ \sec \theta = \sqrt{1 + \tan^2 \theta} \]
Substituting this into the equation gives:
\[ \sqrt{1 + \tan^2 \theta} + \tan \theta = p \]

### Step 3: Isolate the square root
Rearranging the equation, we have:
\[ \sqrt{1 + \tan^2 \theta} = p - \tan \theta \]

### Step 4: Square both sides
Squaring both sides to eliminate the square root results in:
\[ 1 + \tan^2 \theta = (p - \tan \theta)^2 \]

### Step 5: Expand the right-hand side
Expanding the right-hand side using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ 1 + \tan^2 \theta = p^2 - 2p \tan \theta + \tan^2 \theta \]

### Step 6: Simplify the equation
Cancelling \( \tan^2 \theta \) from both sides gives:
\[ 1 = p^2 - 2p \tan \theta \]

### Step 7: Rearrange to isolate \( \tan \theta \)
Rearranging the equation to solve for \( \tan \theta \):
\[ 2p \tan \theta = p^2 - 1 \]
\[ \tan \theta = \frac{p^2 - 1}{2p} \]

### Step 8: Conclusion
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{p^2 - 1}{2p} \]

### Final Answer
The answer corresponds to option (b):
\[ \tan \theta = \frac{p^2 - 1}{2p} \]

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