If `sectheta + tan theta = sqrt(3)`, then the positive value of `sin theta` is:

Options

A. 0
B. `1/2`
C. `sqrt(3)/2`
D. 1

Answer

To solve the problem where \( \sec \theta + \tan \theta = \sqrt{3} \) and find the positive value of \( \sin \theta \), we can follow these steps:

### Step 1: Use the identity for secant and tangent
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
From the problem, we have:
\[ \sec \theta + \tan \theta = \sqrt{3} \]
Substituting this into the identity gives us:
\[ (\sec \theta - \tan \theta)(\sqrt{3}) = 1 \]
Thus,
\[ \sec \theta - \tan \theta = \frac{1}{\sqrt{3}} \]

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

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

### Step 5: Simplify the right-hand side
To simplify \( \sqrt{3} + \frac{1}{\sqrt{3}} \):
\[ \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3 + 1}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]
Thus, we have:
\[ 2 \sec \theta = \frac{4}{\sqrt{3}} \]

### Step 6: Solve for secant
Dividing both sides by 2 gives:
\[ \sec \theta = \frac{2}{\sqrt{3}} \]

### Step 7: Find cosine
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can write:
\[ \frac{1}{\cos \theta} = \frac{2}{\sqrt{3}} \]
Thus,
\[ \cos \theta = \frac{\sqrt{3}}{2} \]

### Step 8: Find sine
Now, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta = \frac{\sqrt{3}}{2} \):
\[ \sin^2 \theta + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 \]
This simplifies to:
\[ \sin^2 \theta + \frac{3}{4} = 1 \]
So,
\[ \sin^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \]
Taking the positive square root (since we want the positive value of \( \sin \theta \)):
\[ \sin \theta = \frac{1}{2} \]

### Final Answer
The positive value of \( \sin \theta \) is:
\[ \boxed{\frac{1}{2}} \]