If `cosec theta-cot theta=(1)/(sqrt3)`,where `theta ne 0` then what is the value of `cos theta`?

Options

A. 0
B. `(sqrt3)/(2)`
C. `1/2`
D. `(1)/(sqrt2)`

Answer

To solve the equation \( \csc \theta - \cot \theta = \frac{1}{\sqrt{3}} \) and find the value of \( \cos \theta \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ \csc \theta - \cot \theta = \frac{1}{\sqrt{3}} \]

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

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

### Step 4: Solve for \( \csc \theta + \cot \theta \)
Rearranging the equation, we find:
\[ \csc \theta + \cot \theta = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Step 5: Set up the system of equations
Now we have two equations:
1. \( \csc \theta - \cot \theta = \frac{1}{\sqrt{3}} \)
2. \( \csc \theta + \cot \theta = \sqrt{3} \)

### Step 6: Add the two equations
Adding these two equations:
\[ (\csc \theta - \cot \theta) + (\csc \theta + \cot \theta) = \frac{1}{\sqrt{3}} + \sqrt{3} \]
This simplifies to:
\[ 2 \csc \theta = \frac{1}{\sqrt{3}} + \sqrt{3} \]

### Step 7: Simplify the right-hand side
To combine the terms on the right, convert \( \sqrt{3} \) to a fraction:
\[ \sqrt{3} = \frac{3}{\sqrt{3}} \]
Thus,
\[ \frac{1}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]
So we have:
\[ 2 \csc \theta = \frac{4}{\sqrt{3}} \]

### Step 8: Solve for \( \csc \theta \)
Dividing both sides by 2:
\[ \csc \theta = \frac{2}{\sqrt{3}} \]

### Step 9: Find \( \sin \theta \)
Since \( \csc \theta = \frac{1}{\sin \theta} \), we have:
\[ \sin \theta = \frac{\sqrt{3}}{2} \]

### Step 10: Find \( \cos \theta \)
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \left(\frac{\sqrt{3}}{2}\right)^2 + \cos^2 \theta = 1 \]
This simplifies to:
\[ \frac{3}{4} + \cos^2 \theta = 1 \]
Subtracting \( \frac{3}{4} \) from both sides gives:
\[ \cos^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 11: Solve for \( \cos \theta \)
Taking the square root:
\[ \cos \theta = \pm \frac{1}{2} \]

### Conclusion
Thus, the value of \( \cos \theta \) is either \( \frac{1}{2} \) or \( -\frac{1}{2} \).