If ` cos theta = - sqrt3/2 and sin prop = - 3/5, " where " theta` does not lie and `prop` lies in the third quadrant, then ` (2 tan prop + sqrt3 tan theta)/(cot^(2) theta + cos prop)` is equal to

Options

A. `7/22`
B. `5/22`
C. `9/22`
D. `22/5`

Explanation

To solve the problem, we need to evaluate the expression:

\[ \frac{2 \tan \delta + \sqrt{3} \tan \theta}{\cot^2 \theta + \cos \delta} \]

Given:
- \(\cos \theta = -\frac{\sqrt{3}}{2}\)
- \(\sin \delta = -\frac{3}{5}\)

### Step 1: Determine \(\theta\) and \(\delta\)

Since \(\cos \theta = -\frac{\sqrt{3}}{2}\), \(\theta\) must lie in the second quadrant (as it cannot lie in the third quadrant).

### Step 2: Find \(\sin \theta\)

Using the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \(\cos^2 \theta\):
\[ \sin^2 \theta + \left(-\frac{\sqrt{3}}{2}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{3}{4} = 1 \]
\[ \sin^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \]
Thus, \(\sin \theta = \frac{1}{2}\) (since \(\theta\) is in the second quadrant).

### Step 3: Find \(\tan \theta\)

Now, we can find \(\tan \theta\):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \]

### Step 4: Determine \(\delta\)

Since \(\sin \delta = -\frac{3}{5}\) and \(\delta\) lies in the third quadrant, we can find \(\cos \delta\) using the Pythagorean identity:
\[ \sin^2 \delta + \cos^2 \delta = 1 \]
\[ \left(-\frac{3}{5}\right)^2 + \cos^2 \delta = 1 \]
\[ \frac{9}{25} + \cos^2 \delta = 1 \]
\[ \cos^2 \delta = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus, \(\cos \delta = -\frac{4}{5}\) (since \(\delta\) is in the third quadrant).

### Step 5: Find \(\tan \delta\)

Now, we can find \(\tan \delta\):
\[ \tan \delta = \frac{\sin \delta}{\cos \delta} = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4} \]

### Step 6: Substitute into the expression

Now we substitute \(\tan \theta\) and \(\tan \delta\) into the expression:
\[ 2 \tan \delta + \sqrt{3} \tan \theta = 2 \cdot \frac{3}{4} + \sqrt{3} \cdot \left(-\frac{1}{\sqrt{3}}\right) \]
\[ = \frac{3}{2} - 1 = \frac{1}{2} \]

Next, we calculate the denominator:
\[ \cot^2 \theta = \frac{1}{\tan^2 \theta} = \frac{1}{\left(-\frac{1}{\sqrt{3}}\right)^2} = 3 \]
Thus,
\[ \cot^2 \theta + \cos \delta = 3 - \frac{4}{5} = 3 - 0.8 = 2.2 = \frac{11}{5} \]

### Step 7: Final calculation

Now we can substitute back into the expression:
\[ \frac{\frac{1}{2}}{\frac{11}{5}} = \frac{1}{2} \cdot \frac{5}{11} = \frac{5}{22} \]

### Final Answer

Thus, the value of the expression is:
\[ \frac{5}{22} \]