If `sec theta + tan theta=(4)/(3)`, then ` sec theta tan theta `=_______

Options

A. `(175)/(24)`
B. `(25)/(576)`
C. `(27)/(576)`
D. `(175)/(576)`

Explanation

To solve the problem where \( \sec \theta + \tan \theta = \frac{4}{3} \) and we need to find \( \sec \theta \tan \theta \), we can follow these steps:

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

### Step 2: Substitute the known value
We substitute \( \sec \theta + \tan \theta = \frac{4}{3} \) into the identity:
\[ \left(\frac{4}{3}\right)(\sec \theta - \tan \theta) = 1 \]

### Step 3: Solve for \( \sec \theta - \tan \theta \)
To find \( \sec \theta - \tan \theta \), we rearrange the equation:
\[ \sec \theta - \tan \theta = \frac{1}{\frac{4}{3}} = \frac{3}{4} \]

### Step 4: Set up the equations
Now we have two equations:
1. \( \sec \theta + \tan \theta = \frac{4}{3} \) (Equation 1)
2. \( \sec \theta - \tan \theta = \frac{3}{4} \) (Equation 2)

### Step 5: Add the equations
Adding Equation 1 and Equation 2:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = \frac{4}{3} + \frac{3}{4} \]
This simplifies to:
\[ 2 \sec \theta = \frac{4}{3} + \frac{3}{4} \]

### Step 6: Find a common denominator and simplify
The common denominator for \( \frac{4}{3} \) and \( \frac{3}{4} \) is 12:
\[ \frac{4}{3} = \frac{16}{12}, \quad \frac{3}{4} = \frac{9}{12} \]
Thus:
\[ 2 \sec \theta = \frac{16}{12} + \frac{9}{12} = \frac{25}{12} \]
So:
\[ \sec \theta = \frac{25}{24} \]

### Step 7: Subtract the equations
Now we subtract Equation 2 from Equation 1:
\[ (\sec \theta + \tan \theta) - (\sec \theta - \tan \theta) = \frac{4}{3} - \frac{3}{4} \]
This simplifies to:
\[ 2 \tan \theta = \frac{4}{3} - \frac{3}{4} \]

### Step 8: Find a common denominator and simplify
Using the common denominator of 12 again:
\[ \frac{4}{3} = \frac{16}{12}, \quad \frac{3}{4} = \frac{9}{12} \]
Thus:
\[ 2 \tan \theta = \frac{16}{12} - \frac{9}{12} = \frac{7}{12} \]
So:
\[ \tan \theta = \frac{7}{24} \]

### Step 9: Calculate \( \sec \theta \tan \theta \)
Now we can find \( \sec \theta \tan \theta \):
\[ \sec \theta \tan \theta = \left(\frac{25}{24}\right) \left(\frac{7}{24}\right) = \frac{175}{576} \]

### Final Answer
Thus, the value of \( \sec \theta \tan \theta \) is:
\[ \sec \theta \tan \theta = \frac{175}{576} \]