A is an angle in the fourth quadrant. If satisfies the trigonometric equation `3(3-tan^2" "A-cot A)^2` = 1. Which one of the following is a value of A?

Options

A. `300^@`
B. `315^@`
C. `330^@`
D. `345^@`

Answer

To solve the equation \( 3(3 - \tan^2 A - \cot A)^2 = 1 \) for an angle \( A \) in the fourth quadrant, we can follow these steps:

### Step 1: Simplify the equation
Start by dividing both sides of the equation by 3:
\[ (3 - \tan^2 A - \cot A)^2 = \frac{1}{3} \]

### Step 2: Take the square root
Taking the square root of both sides gives us two cases:
\[ 3 - \tan^2 A - \cot A = \frac{1}{\sqrt{3}} \quad \text{or} \quad 3 - \tan^2 A - \cot A = -\frac{1}{\sqrt{3}} \]

### Step 3: Solve for \( \tan A \) and \( \cot A \)
Let's focus on the first case:
\[ 3 - \tan^2 A - \cot A = \frac{1}{\sqrt{3}} \]
Rearranging gives:
\[ \tan^2 A + \cot A = 3 - \frac{1}{\sqrt{3}} \]

Now, using the identity \( \cot A = \frac{1}{\tan A} \), we can substitute:
\[ \tan^2 A + \frac{1}{\tan A} = 3 - \frac{1}{\sqrt{3}} \]

Let \( x = \tan A \). Then we have:
\[ x^2 + \frac{1}{x} = 3 - \frac{1}{\sqrt{3}} \]

### Step 4: Multiply through by \( x \)
To eliminate the fraction, multiply through by \( x \):
\[ x^3 + 1 = x(3 - \frac{1}{\sqrt{3}}) \]
This simplifies to:
\[ x^3 - (3 - \frac{1}{\sqrt{3}})x + 1 = 0 \]

### Step 5: Check possible angles in the fourth quadrant
Now we can check the possible angles in the fourth quadrant. The angle \( A \) in the fourth quadrant could be \( 300^\circ \) or \( -60^\circ \).

### Step 6: Calculate \( \tan A \) and \( \cot A \) for \( A = 300^\circ \)
For \( A = 300^\circ \):
\[ \tan 300^\circ = \tan(-60^\circ) = -\tan 60^\circ = -\sqrt{3} \]
\[ \cot 300^\circ = \cot(-60^\circ) = -\cot 60^\circ = -\frac{1}{\sqrt{3}} \]

### Step 7: Substitute back into the equation
Substituting these values back into the equation:
\[ 3 - (-\sqrt{3})^2 - \left(-\frac{1}{\sqrt{3}}\right) = 3 - 3 + \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Step 8: Verify the equality
Now we can check if:
\[ 3(3 - \tan^2 300^\circ - \cot 300^\circ)^2 = 1 \]
This holds true, confirming that \( A = 300^\circ \) is indeed a solution.

### Conclusion
Thus, the value of \( A \) that satisfies the equation is:
\[ \boxed{300^\circ} \]