If `"cosec "theta=2`, then the value of `tan theta` will be :

Options

A. `(sqrt3)/(2)`
B. `(1)/(sqrt3)`
C. `(1)/(2)`
D. `(2)/(sqrt3)`.

Answer

To solve the problem where \( \csc \theta = 2 \) and we need to find the value of \( \tan \theta \), we can follow these steps:

### Step 1: Understand the relationship between cosecant and sine
We know that:
\[ \csc \theta = \frac{1}{\sin \theta} \]
Given that \( \csc \theta = 2 \), we can find \( \sin \theta \):
\[ \sin \theta = \frac{1}{\csc \theta} = \frac{1}{2} \]

### Step 2: Use the Pythagorean identity
We can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta \):
\[ \left(\frac{1}{2}\right)^2 + \cos^2 \theta = 1 \]
This simplifies to:
\[ \frac{1}{4} + \cos^2 \theta = 1 \]

### Step 3: Solve for \( \cos^2 \theta \)
Rearranging the equation gives:
\[ \cos^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \]
Taking the square root gives:
\[ \cos \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \quad \text{(considering the principal value)} \]

### Step 4: Find \( \tan \theta \)
Now, we can find \( \tan \theta \) using the definition:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting the values we found:
\[ \tan \theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{1}{\sqrt{3}} \]