In triangle `A B C ,` if `sinAcosB=1/4 and 3t a n A=t a n B ,t h e ncot^2A` is equal to

Options

A. 2
B. 3
C. 4
D. 5

Explanation

Let's solve the problem step by step.

### Given:
1. \( \sin A \cos B = \frac{1}{4} \)
2. \( 3 \tan A = \tan B \)

We need to find \( \cot^2 A \).

### Step 1: Express \( \tan B \) in terms of \( \tan A \)
From the second equation, we can express \( \tan B \) as:
\[ \tan B = 3 \tan A \]

### Step 2: Rewrite \( \tan A \) and \( \tan B \) in terms of sine and cosine
Using the definitions of tangent:
\[ \tan A = \frac{\sin A}{\cos A}, \quad \tan B = \frac{\sin B}{\cos B} \]
Substituting these into the equation from Step 1:
\[ \frac{\sin B}{\cos B} = 3 \cdot \frac{\sin A}{\cos A} \]
Cross-multiplying gives:
\[ \sin B \cos A = 3 \sin A \cos B \]

### Step 3: Substitute \( \sin A \cos B \) from the given condition
We know from the problem statement that:
\[ \sin A \cos B = \frac{1}{4} \]
Substituting this into the equation from Step 2:
\[ \sin B \cos A = 3 \cdot \frac{1}{4} \]
This simplifies to:
\[ \sin B \cos A = \frac{3}{4} \]

### Step 4: Use the sine addition formula
Now we have:
\[ \sin A \cos B + \sin B \cos A = \frac{1}{4} + \frac{3}{4} = 1 \]
This expression can be recognized as:
\[ \sin(A + B) = 1 \]
Thus, we have:
\[ A + B = 90^\circ \]

### Step 5: Find angle \( C \)
In triangle \( ABC \), the sum of angles is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
Since \( A + B = 90^\circ \):
\[ C = 180^\circ - 90^\circ = 90^\circ \]

### Step 6: Relate \( \tan B \) to \( \cot A \)
From the identity \( \tan(90^\circ - A) = \cot A \), we have:
\[ \tan B = \tan(90^\circ - A) = \cot A \]
Thus, we can write:
\[ 3 \tan A = \cot A \]

### Step 7: Convert to cotangent
Using the identity \( \cot A = \frac{1}{\tan A} \):
\[ 3 \tan A = \frac{1}{\tan A} \]
Multiplying both sides by \( \tan A \) (assuming \( \tan A \neq 0 \)):
\[ 3 \tan^2 A = 1 \]
This gives:
\[ \tan^2 A = \frac{1}{3} \]

### Step 8: Find \( \cot^2 A \)
Since \( \cot A = \frac{1}{\tan A} \):
\[ \cot^2 A = \frac{1}{\tan^2 A} = \frac{1}{\frac{1}{3}} = 3 \]

### Final Answer
Thus, the value of \( \cot^2 A \) is:
\[ \cot^2 A = 3 \]