If `secA=-2` and A lies in third quadrant, find the value of `(4 cot^(2)A-3sin^(2)A)`.

Options

A. `11/12`
B. `-11/12`
C. 1
D. None of these

Explanation

To solve the problem step by step, we need to find the value of \( 4 \cot^2 A - 3 \sin^2 A \) given that \( \sec A = -2 \) and \( A \) lies in the third quadrant.

### Step 1: Find \( \cos A \)
Since \( \sec A = \frac{1}{\cos A} \), we can find \( \cos A \):
\[ \cos A = \frac{1}{\sec A} = \frac{1}{-2} = -\frac{1}{2} \]

**Hint**: Remember that secant is the reciprocal of cosine.

### Step 2: Find \( \sin^2 A \)
Using the Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \), we can find \( \sin^2 A \):
\[ \sin^2 A = 1 - \cos^2 A = 1 - \left(-\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4} \]

**Hint**: Always check the quadrant to determine the sign of sine.

### Step 3: Find \( \sin A \)
Since \( A \) is in the third quadrant, where sine is negative:
\[ \sin A = -\sqrt{\sin^2 A} = -\sqrt{\frac{3}{4}} = -\frac{\sqrt{3}}{2} \]

**Hint**: The sine function is negative in the third quadrant.

### Step 4: Find \( \cot A \)
Using the definition of cotangent:
\[ \cot A = \frac{\cos A}{\sin A} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

**Hint**: Cotangent is the ratio of cosine to sine.

### Step 5: Calculate \( \cot^2 A \) and \( \sin^2 A \)
Now we calculate \( \cot^2 A \):
\[ \cot^2 A = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 6: Substitute values into the expression
Now we can substitute \( \cot^2 A \) and \( \sin^2 A \) into the expression \( 4 \cot^2 A - 3 \sin^2 A \):
\[ 4 \cot^2 A - 3 \sin^2 A = 4 \left(\frac{1}{3}\right) - 3 \left(\frac{3}{4}\right) \]
Calculating each term:
\[ = \frac{4}{3} - \frac{9}{4} \]

### Step 7: Find a common denominator and simplify
The common denominator for \( \frac{4}{3} \) and \( \frac{9}{4} \) is 12:
\[ = \frac{4 \times 4}{12} - \frac{9 \times 3}{12} = \frac{16}{12} - \frac{27}{12} = \frac{16 - 27}{12} = \frac{-11}{12} \]

### Final Result
Thus, the value of \( 4 \cot^2 A - 3 \sin^2 A \) is:
\[ \boxed{-\frac{11}{12}} \]