If `x=15^(@)`, evaluate : `8sin2xcos4xsin6x`

Answer

To evaluate the expression \( 8 \sin 2x \cos 4x \sin 6x \) given \( x = 15^\circ \), we can follow these steps:

### Step 1: Substitute the value of \( x \)
We start by substituting \( x = 15^\circ \) into the expression.

\[ 2x = 2 \times 15^\circ = 30^\circ \]
\[ 4x = 4 \times 15^\circ = 60^\circ \]
\[ 6x = 6 \times 15^\circ = 90^\circ \]

Now we rewrite the expression with these values:

\[ 8 \sin 30^\circ \cos 60^\circ \sin 90^\circ \]

### Step 2: Use trigonometric values
Next, we use the known values of the trigonometric functions:

\[ \sin 30^\circ = \frac{1}{2} \]
\[ \cos 60^\circ = \frac{1}{2} \]
\[ \sin 90^\circ = 1 \]

### Step 3: Substitute the trigonometric values into the expression
Now we substitute these values back into the expression:

\[ 8 \times \frac{1}{2} \times \frac{1}{2} \times 1 \]

### Step 4: Simplify the expression
Now we simplify the expression step by step:

\[ = 8 \times \frac{1}{2} \times \frac{1}{2} \times 1 \]
\[ = 8 \times \frac{1}{4} \]
\[ = 2 \]

### Final Answer
Thus, the final answer is:

\[ \boxed{2} \]

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