If `"cosec"theta - cot theta = 1/3`, then the value of `"cosec"theta + cot theta` is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem where we need to find the value of \( \csc \theta + \cot \theta \) given that \( \csc \theta - \cot \theta = \frac{1}{3} \), we can follow these steps:

### Step 1: Use the identity for the difference of squares
We know the identity:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
This can be expressed as:
\[ (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) = 1 \]

### Step 2: Substitute the given value
From the problem, we have:
\[ \csc \theta - \cot \theta = \frac{1}{3} \]
Let \( x = \csc \theta + \cot \theta \). Substituting this into the identity gives us:
\[ \left(\frac{1}{3}\right)(x) = 1 \]

### Step 3: Solve for \( x \)
To find \( x \), we can rearrange the equation:
\[ x = \frac{1}{1/3} = 3 \]

### Conclusion
Thus, the value of \( \csc \theta + \cot \theta \) is:
\[ \boxed{3} \]