If `tantheta-cottheta=cosectheta, 0^(@) lt theta lt 90^(@)`, then what is the value of `(2tantheta-costheta)/(sqrt(3)cottheta+sectheta)`?

Options

A. `(2sqrt(3)-1)/3`
B. `(23(2sqrt(3)-1))/3`
C. `(4sqrt(3)-1)/6`
D. `(3sqrt(3)-1)/6`

Answer

To solve the problem, we start with the given equation:

\[ \tan \theta - \cot \theta = \csc \theta \]

### Step 1: Rewrite the trigonometric functions
We can express \(\tan \theta\), \(\cot \theta\), and \(\csc \theta\) in terms of sine and cosine:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \csc \theta = \frac{1}{\sin \theta} \]

Substituting these into the equation gives:

\[ \frac{\sin \theta}{\cos \theta} - \frac{\cos \theta}{\sin \theta} = \frac{1}{\sin \theta} \]

### Step 2: Find a common denominator
The left-hand side can be simplified by finding a common denominator:

\[ \frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta} \]

### Step 3: Cross-multiply
Cross-multiplying gives:

\[ (\sin^2 \theta - \cos^2 \theta) = \frac{\sin \theta \cos \theta}{\sin \theta} \]

This simplifies to:

\[ \sin^2 \theta - \cos^2 \theta = \cos \theta \]

### Step 4: Rearrange the equation
Rearranging the equation gives:

\[ \sin^2 \theta - \cos^2 \theta - \cos \theta = 0 \]

### Step 5: Substitute \(\sin^2 \theta\)
Using the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\):

\[ (1 - \cos^2 \theta) - \cos^2 \theta - \cos \theta = 0 \]

This simplifies to:

\[ 1 - 2\cos^2 \theta - \cos \theta = 0 \]

### Step 6: Rearranging into a standard quadratic form
Rearranging gives:

\[ 2\cos^2 \theta + \cos \theta - 1 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here, \(a = 2\), \(b = 1\), and \(c = -1\):

\[ \cos \theta = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
\[ = \frac{-1 \pm \sqrt{1 + 8}}{4} \]
\[ = \frac{-1 \pm 3}{4} \]

Calculating the two possible values:

1. \(\cos \theta = \frac{2}{4} = \frac{1}{2}\) (valid since \(0 < \theta < 90\))
2. \(\cos \theta = \frac{-4}{4} = -1\) (not valid)

Thus, \(\cos \theta = \frac{1}{2}\) implies:

\[ \theta = 60^\circ \]

### Step 8: Calculate \(\tan \theta\) and \(\cot \theta\)
Now, we can find \(\tan \theta\) and \(\cot \theta\):

\[ \tan 60^\circ = \sqrt{3}, \quad \cot 60^\circ = \frac{1}{\sqrt{3}} \]

### Step 9: Substitute into the expression
Now we substitute into the expression we need to evaluate:

\[ \frac{2\tan \theta - \cos \theta}{\sqrt{3}\cot \theta + \sec \theta} \]

Calculating each component:

- \(2\tan 60^\circ = 2\sqrt{3}\)
- \(\cos 60^\circ = \frac{1}{2}\)
- \(\sqrt{3}\cot 60^\circ = \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1\)
- \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = 2\)

Thus, the expression becomes:

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

### Step 10: Simplify the numerator
The numerator can be simplified:

\[ 2\sqrt{3} - \frac{1}{2} = \frac{4\sqrt{3} - 1}{2} \]

So the final expression is:

\[ \frac{\frac{4\sqrt{3} - 1}{2}}{3} = \frac{4\sqrt{3} - 1}{6} \]

### Final Result
Thus, the value of the expression is:

\[ \frac{4\sqrt{3} - 1}{6} \]