If `cosectheta-cottheta=1/5`, then the value of `cosectheta+cottheta`.

Options

A. `1/5`
B. `1/25`
C. `(-1)/25`
D. 5

Answer

To solve the equation \( \csc \theta - \cot \theta = \frac{1}{5} \) and find the value of \( \csc \theta + \cot \theta \), we can follow these steps:

### Step 1: Set up the equations
Let:
\[ x = \csc \theta + \cot \theta \]
\[ y = \csc \theta - \cot \theta \]
From the problem, we know:
\[ y = \frac{1}{5} \]

### Step 2: Use the identity
We know the identity:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
This can be factored as:
\[ (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) = 1 \]
Substituting our variables, we have:
\[ y \cdot x = 1 \]

### Step 3: Substitute the known value
Substituting \( y = \frac{1}{5} \) into the equation:
\[ \frac{1}{5} \cdot x = 1 \]

### Step 4: Solve for \( x \)
To find \( x \), multiply both sides by 5:
\[ x = 5 \]

### Conclusion
Thus, the value of \( \csc \theta + \cot \theta \) is:
\[ \csc \theta + \cot \theta = 5 \]