If sec A+tan A=x, then tan A:

Options

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

Answer

To solve the problem where sec A + tan A = x and we need to find tan A, we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \sec A + \tan A = x \]

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

### Step 3: Substitute the known value
Since we know that \(\sec A + \tan A = x\), we can substitute this into our identity:
\[ (\sec A - \tan A)(x) = 1 \]

### Step 4: Solve for \(\sec A - \tan A\)
Now, we can isolate \(\sec A - \tan A\):
\[ \sec A - \tan A = \frac{1}{x} \]

### Step 5: Add the two equations
Now we have two equations:
1. \(\sec A + \tan A = x\)
2. \(\sec A - \tan A = \frac{1}{x}\)

We can add these two equations:
\[ (\sec A + \tan A) + (\sec A - \tan A) = x + \frac{1}{x} \]
This simplifies to:
\[ 2\sec A = x + \frac{1}{x} \]

### Step 6: Solve for \(\sec A\)
Now, divide both sides by 2:
\[ \sec A = \frac{x + \frac{1}{x}}{2} \]

### Step 7: Find \(\tan A\)
Now, we can find \(\tan A\) using the first equation:
\[ \tan A = x - \sec A \]
Substituting \(\sec A\):
\[ \tan A = x - \frac{x + \frac{1}{x}}{2} \]

### Step 8: Simplify the expression
Now, simplify the right side:
\[ \tan A = x - \left(\frac{2x + \frac{1}{x}}{2}\right) \]
\[ \tan A = \frac{2x - (2x + \frac{1}{x})}{2} \]
\[ \tan A = \frac{-\frac{1}{x}}{2} \]
\[ \tan A = -\frac{1}{2x} \]

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