If `"cosec"theta-cottheta=(1)/(sqrt(3))` where `thetane0`, then what is the value of `costheta` ?

Options

A. 0
B. `(sqrt(3))/(2)`
C. `(1)/(2)`
D. `(1)/(sqrt(2))`

Explanation

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 equation in terms of sine and cosine
We know that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Substituting these into the equation gives:
\[ \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} = \frac{1}{\sqrt{3}} \]
This simplifies to:
\[ \frac{1 - \cos \theta}{\sin \theta} = \frac{1}{\sqrt{3}} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying yields:
\[ 1 - \cos \theta = \frac{\sin \theta}{\sqrt{3}} \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (1 - \cos \theta)^2 = \left(\frac{\sin \theta}{\sqrt{3}}\right)^2 \]
Expanding both sides:
\[ 1 - 2\cos \theta + \cos^2 \theta = \frac{\sin^2 \theta}{3} \]

### Step 4: Use the Pythagorean identity
Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \), so we can substitute \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ 1 - 2\cos \theta + \cos^2 \theta = \frac{1 - \cos^2 \theta}{3} \]

### Step 5: Multiply through by 3 to eliminate the fraction
Multiplying through by 3 gives:
\[ 3(1 - 2\cos \theta + \cos^2 \theta) = 1 - \cos^2 \theta \]
Expanding this:
\[ 3 - 6\cos \theta + 3\cos^2 \theta = 1 - \cos^2 \theta \]

### Step 6: Rearrange the equation
Bringing all terms to one side:
\[ 3\cos^2 \theta + \cos^2 \theta - 6\cos \theta + 3 - 1 = 0 \]
This simplifies to:
\[ 4\cos^2 \theta - 6\cos \theta + 2 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4, b = -6, c = 2 \):
\[ \cos \theta = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 4 \cdot 2}}{2 \cdot 4} \]
Calculating the discriminant:
\[ \sqrt{36 - 32} = \sqrt{4} = 2 \]
Thus:
\[ \cos \theta = \frac{6 \pm 2}{8} \]
This gives us two solutions:
\[ \cos \theta = \frac{8}{8} = 1 \quad \text{or} \quad \cos \theta = \frac{4}{8} = \frac{1}{2} \]

### Step 8: Determine valid solutions
Since \( \theta \neq 0 \), we discard \( \cos \theta = 1 \) and keep:
\[ \cos \theta = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \boxed{\frac{1}{2}} \]