If `(1)/(cosectheta+cottheta)-cosectheta-tantheta=3ksecthetacosectheta`, then value of k.

Options

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

Answer

To solve the equation

\[ \frac{1}{\csc \theta + \cot \theta} - \csc \theta - \tan \theta = 3k \sec \theta \csc \theta, \]

we will follow these steps:

### Step 1: Substitute values for trigonometric functions
Let's take \(\theta = 45^\circ\) to simplify our calculations. We know:
- \(\csc 45^\circ = \sqrt{2}\)
- \(\cot 45^\circ = 1\)
- \(\tan 45^\circ = 1\)
- \(\sec 45^\circ = \sqrt{2}\)

### Step 2: Substitute into the equation
Substituting these values into the equation gives us:

\[ \frac{1}{\sqrt{2} + 1} - \sqrt{2} - 1 = 3k \cdot \sqrt{2} \cdot \sqrt{2}. \]

### Step 3: Simplify the left side
First, we simplify the left side:

\[ \frac{1}{\sqrt{2} + 1} = \frac{1}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1. \]

So we have:

\[ \sqrt{2} - 1 - \sqrt{2} - 1 = 3k \cdot 2. \]

### Step 4: Combine like terms
Now, combining like terms on the left side:

\[ (\sqrt{2} - \sqrt{2}) - 1 - 1 = 3k \cdot 2, \]

which simplifies to:

\[ -2 = 6k. \]

### Step 5: Solve for \(k\)
Now, we can solve for \(k\):

\[ k = \frac{-2}{6} = -\frac{1}{3}. \]

### Conclusion
Thus, the value of \(k\) is

\[ \boxed{-\frac{1}{3}}. \]