If `cosec theta - cottheta = (1)/(3)` , then `cosec theta + cot theta` =

Options

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

Answer

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

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

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

### Step 3: Solve for \( x \)
To isolate \( x \), multiply both sides by 3:
\[ x = 3 \]

### Conclusion
Thus, we find that:
\[ \csc \theta + \cot \theta = 3 \]