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

Options

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

Answer

To find the value of sin θ given that sec θ + tan θ = p, we can follow these steps:

### Step 1: Use the identity
We know that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 2: Substitute the given value
Since we are given that:
\[ \sec \theta + \tan \theta = p \]
We can substitute this into our identity:
\[ (\sec \theta - \tan \theta)(p) = 1 \]
From this, we can solve for \(\sec \theta - \tan \theta\):
\[ \sec \theta - \tan \theta = \frac{1}{p} \]

### Step 3: Set up the equations
Now we have two equations:
1. \(\sec \theta + \tan \theta = p\)
2. \(\sec \theta - \tan \theta = \frac{1}{p}\)

### Step 4: Add the equations
Adding these two equations:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = p + \frac{1}{p} \]
This simplifies to:
\[ 2 \sec \theta = p + \frac{1}{p} \]
Thus, we can find \(\sec \theta\):
\[ \sec \theta = \frac{p + \frac{1}{p}}{2} \]

### Step 5: Find cos θ
Since \(\sec \theta = \frac{1}{\cos \theta}\), we have:
\[ \cos \theta = \frac{2}{p + \frac{1}{p}} = \frac{2p}{p^2 + 1} \]

### Step 6: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting \(\cos^2 \theta\):
\[ \sin^2 \theta = 1 - \left(\frac{2p}{p^2 + 1}\right)^2 \]

### Step 7: Simplify sin² θ
Calculating \(\cos^2 \theta\):
\[ \cos^2 \theta = \frac{4p^2}{(p^2 + 1)^2} \]
So,
\[ \sin^2 \theta = 1 - \frac{4p^2}{(p^2 + 1)^2} = \frac{(p^2 + 1)^2 - 4p^2}{(p^2 + 1)^2} \]
Expanding the numerator:
\[ (p^2 + 1)^2 - 4p^2 = p^4 + 2p^2 + 1 - 4p^2 = p^4 - 2p^2 + 1 = (p^2 - 1)^2 \]
Thus,
\[ \sin^2 \theta = \frac{(p^2 - 1)^2}{(p^2 + 1)^2} \]

### Step 8: Take the square root
Taking the square root gives us:
\[ \sin \theta = \frac{p^2 - 1}{p^2 + 1} \]

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