If tan `theta + sec theta`=4 , the value of sin `theta cos theta`.

Answer

To solve the problem where \( \tan \theta + \sec \theta = 4 \), we need to find the value of \( \sin \theta \cos \theta \). Let's go through the steps systematically:

### Step 1: Set up the equations
We know that:
\[ \tan \theta + \sec \theta = 4 \]
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \), we can express this as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 2: Substitute the known value
Let:
\[ x = \sec \theta + \tan \theta \]
From the problem, we know:
\[ x = 4 \]
Thus, we can write:
\[ \sec \theta - \tan \theta = \frac{1}{x} = \frac{1}{4} \]

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

Adding these two equations:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = 4 + \frac{1}{4} \]
This simplifies to:
\[ 2 \sec \theta = 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]
Thus:
\[ \sec \theta = \frac{17}{8} \]

### Step 4: Find \( \cos \theta \)
Since \( \sec \theta = \frac{1}{\cos \theta} \), we have:
\[ \cos \theta = \frac{8}{17} \]

### Step 5: Find \( \sin \theta \)
Using the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
We can find \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{8}{17}\right)^2 \]
Calculating \( \cos^2 \theta \):
\[ \cos^2 \theta = \frac{64}{289} \]
Thus:
\[ \sin^2 \theta = 1 - \frac{64}{289} = \frac{289 - 64}{289} = \frac{225}{289} \]
Taking the square root gives:
\[ \sin \theta = \frac{15}{17} \]

### Step 6: Calculate \( \sin \theta \cos \theta \)
Now we can find \( \sin \theta \cos \theta \):
\[ \sin \theta \cos \theta = \left(\frac{15}{17}\right) \left(\frac{8}{17}\right) = \frac{120}{289} \]

### Final Answer
Thus, the value of \( \sin \theta \cos \theta \) is:
\[ \boxed{\frac{120}{289}} \]