`12sqrt(3)-(8cos x)((3sqrt(3))/(2)cos x)=`

Options

A. `sin^(2)x`
B. `12sqrt(3)sin^(2)x`
C. `12sqrt(3)-12sqrt(3)cos x`
D. `12sqrt(3)cos^(2)x`

Explanation

To solve the expression \( 12\sqrt{3} - (8\cos x)\left(\frac{3\sqrt{3}}{2}\cos x\right) \), we will follow these steps:

### Step 1: Simplify the expression
We start with the expression:
\[ 12\sqrt{3} - (8\cos x)\left(\frac{3\sqrt{3}}{2}\cos x\right) \]

### Step 2: Multiply the terms in the second part
Now, we need to multiply \( 8\cos x \) and \( \frac{3\sqrt{3}}{2}\cos x \):
\[ (8\cos x)\left(\frac{3\sqrt{3}}{2}\cos x\right) = 8 \cdot \frac{3\sqrt{3}}{2} \cdot \cos^2 x \]
Calculating the constant:
\[ 8 \cdot \frac{3\sqrt{3}}{2} = 12\sqrt{3} \]
Thus, we have:
\[ 12\sqrt{3}\cos^2 x \]

### Step 3: Substitute back into the expression
Now we can substitute this back into the original expression:
\[ 12\sqrt{3} - 12\sqrt{3}\cos^2 x \]

### Step 4: Factor out the common term
We can factor out \( 12\sqrt{3} \):
\[ 12\sqrt{3}(1 - \cos^2 x) \]

### Step 5: Use the trigonometric identity
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can replace \( 1 - \cos^2 x \) with \( \sin^2 x \):
\[ 12\sqrt{3}(1 - \cos^2 x) = 12\sqrt{3}\sin^2 x \]

### Final Result
Thus, the simplified expression is:
\[ 12\sqrt{3}\sin^2 x \]