If ` x= 15 ^(@) ,` evaluate :
` 4 cos 2x . Sin 4x. tan 3x- 1`

Answer

To evaluate the expression \( 4 \cos(2x) \cdot \sin(4x) \cdot \tan(3x) - 1 \) given that \( x = 15^\circ \), we will follow these steps:

### Step 1: Substitute the value of \( x \)
Given \( x = 15^\circ \), we substitute this into the expression:
\[ 4 \cos(2 \cdot 15^\circ) \cdot \sin(4 \cdot 15^\circ) \cdot \tan(3 \cdot 15^\circ) - 1 \]

### Step 2: Calculate the angles
Now calculate the angles:
- \( 2 \cdot 15^\circ = 30^\circ \)
- \( 4 \cdot 15^\circ = 60^\circ \)
- \( 3 \cdot 15^\circ = 45^\circ \)

So, we rewrite the expression:
\[ 4 \cos(30^\circ) \cdot \sin(60^\circ) \cdot \tan(45^\circ) - 1 \]

### Step 3: Find the trigonometric values
Now we will find the values of the trigonometric functions:
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \tan(45^\circ) = 1 \)

### Step 4: Substitute the trigonometric values
Substituting these values back into the expression gives:
\[ 4 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cdot 1 - 1 \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ 4 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = 4 \cdot \frac{3}{4} = 3 \]
Thus, we have:
\[ 3 - 1 = 2 \]

### Final Answer
The final answer is:
\[ \boxed{2} \]