If `Cos theta = sqrt3//2` , then find the value of `(2 sin^2 theta)/((1-cot^2 theta)) + (sec^2 theta + cosec theta)`

Options

A. 29/12
B. 43/12
C. 37/12
D. 19/12

Answer

To solve the problem, we need to find the value of the expression given that \( \cos \theta = \frac{\sqrt{3}}{2} \).

### Step 1: Find \( \sin^2 \theta \)
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the value of \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 \]
Calculating \( \left(\frac{\sqrt{3}}{2}\right)^2 \):
\[ \sin^2 \theta + \frac{3}{4} = 1 \]
Now, solving for \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 2: Find \( \cot^2 \theta \)
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Substituting the values we have:
\[ \cot \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]
Now, calculating \( \cot^2 \theta \):
\[ \cot^2 \theta = (\sqrt{3})^2 = 3 \]

### Step 3: Substitute values into the expression
We need to evaluate:
\[ \frac{2 \sin^2 \theta}{1 - \cot^2 \theta} + \sec^2 \theta + \csc \theta \]
Substituting \( \sin^2 \theta \) and \( \cot^2 \theta \):
\[ \frac{2 \cdot \frac{1}{4}}{1 - 3} + \sec^2 \theta + \csc \theta \]
Calculating the denominator:
\[ 1 - 3 = -2 \]
Now substituting:
\[ \frac{\frac{1}{2}}{-2} + \sec^2 \theta + \csc \theta = -\frac{1}{4} + \sec^2 \theta + \csc \theta \]

### Step 4: Find \( \sec^2 \theta \) and \( \csc \theta \)
Calculating \( \sec^2 \theta \):
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \quad \Rightarrow \quad \sec^2 \theta = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]

Calculating \( \csc \theta \):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 5: Substitute \( \sec^2 \theta \) and \( \csc \theta \) back into the expression
Now substituting these values:
\[ -\frac{1}{4} + \frac{4}{3} + 2 \]
Converting \( 2 \) to a fraction:
\[ 2 = \frac{6}{3} \]
Now combining:
\[ -\frac{1}{4} + \frac{4}{3} + \frac{6}{3} = -\frac{1}{4} + \frac{10}{3} \]

### Step 6: Find a common denominator
The common denominator between 4 and 3 is 12:
\[ -\frac{1}{4} = -\frac{3}{12}, \quad \frac{10}{3} = \frac{40}{12} \]
Now combining:
\[ -\frac{3}{12} + \frac{40}{12} = \frac{37}{12} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{37}{12} \]