If `secA+tanA=p` , then what is the values of sin A?

Options

A. `(p^(2)-1)/(p^(2)+1)`
B. `(p^(2)+1)/(p^(2)-1)`
C. 1
D. None of these

Explanation

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

### Step 1: Rewrite the equation in terms of sine and cosine
We know that:
\[ \sec A = \frac{1}{\cos A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{\cos A} + \frac{\sin A}{\cos A} = p \]
Combining the fractions gives:
\[ \frac{1 + \sin A}{\cos A} = p \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 1 + \sin A = p \cos A \]

### Step 3: Square both sides
To eliminate the cosine, we can square both sides:
\[ (1 + \sin A)^2 = (p \cos A)^2 \]
Expanding both sides results in:
\[ 1 + 2\sin A + \sin^2 A = p^2 \cos^2 A \]

### Step 4: Substitute \( \cos^2 A \) using the Pythagorean identity
Using the identity \( \cos^2 A = 1 - \sin^2 A \), we can substitute:
\[ 1 + 2\sin A + \sin^2 A = p^2 (1 - \sin^2 A) \]
Expanding the right side gives:
\[ 1 + 2\sin A + \sin^2 A = p^2 - p^2 \sin^2 A \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:
\[ \sin^2 A + p^2 \sin^2 A + 2\sin A + 1 - p^2 = 0 \]
This simplifies to:
\[ (1 + p^2) \sin^2 A + 2\sin A + (1 - p^2) = 0 \]

### Step 6: Use the quadratic formula
This is a quadratic equation in terms of \( \sin A \). We can apply the quadratic formula:
\[ \sin A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 + p^2 \), \( b = 2 \), and \( c = 1 - p^2 \). Thus:
\[ \sin A = \frac{-2 \pm \sqrt{2^2 - 4(1 + p^2)(1 - p^2)}}{2(1 + p^2)} \]

### Step 7: Simplify the expression under the square root
Calculating the discriminant:
\[ \sqrt{4 - 4(1 + p^2)(1 - p^2)} = \sqrt{4 - 4(1 - p^4)} = \sqrt{4p^4} \]
This simplifies to:
\[ 2p^2 \]

### Step 8: Substitute back into the formula
Now substituting back into the formula gives:
\[ \sin A = \frac{-2 \pm 2p^2}{2(1 + p^2)} = \frac{-1 \pm p^2}{1 + p^2} \]

### Step 9: Choose the correct root
Since \( \sin A \) must be between -1 and 1, we choose:
\[ \sin A = \frac{p^2 - 1}{p^2 + 1} \]

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

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