What is the value of `((32cos^(6)x-48cos^(4)x+18cos^(2)x-1))/(4sinxcosxsin(60-x)cos(60-x)sin(60+x)cos(60+x))`?

Options

A. 4 tan 6x
B. 4 cot 6x
C. 8 cot 6x
D. 8 tan 6x

Answer

To solve the expression

\[ \frac{32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1}{4\sin x \cos x \sin(60 - x) \cos(60 - x) \sin(60 + x) \cos(60 + x)}, \]

we will follow these steps:

### Step 1: Simplify the Numerator

We can factor out a common factor from the numerator. Notice that we can factor out 2:

\[ 32\cos^6 x - 48\cos^4 x + 18\cos^2 x - 1 = 2(16\cos^6 x - 24\cos^4 x + 9\cos^2 x - \frac{1}{2}). \]

### Step 2: Rewrite the Denominator

Next, we simplify the denominator. We can use the identities for sine and cosine:

\[ \sin(60 - x) = \sin 60 \cos x - \cos 60 \sin x = \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x, \]
\[ \cos(60 - x) = \cos 60 \cos x + \sin 60 \sin x = \frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x. \]

Thus, we can express the denominator as:

\[ 4 \sin x \cos x \sin(60 - x) \cos(60 - x) \sin(60 + x) \cos(60 + x). \]

### Step 3: Use Product-to-Sum Identities

Using the product-to-sum identities, we can simplify the sine and cosine products in the denominator.

### Step 4: Factor the Numerator

The numerator can be recognized as a polynomial in terms of \( \cos^2 x \). We can let \( y = \cos^2 x \):

\[ 16y^3 - 24y^2 + 9y - \frac{1}{2}. \]

### Step 5: Find Roots of the Polynomial

We can find the roots of the polynomial to factor it further.

### Step 6: Substitute Back and Simplify

After factoring and simplifying, we can substitute back \( y = \cos^2 x \) to express everything in terms of \( \cos x \).

### Step 7: Final Simplification

Finally, we will simplify the entire expression and look for any common factors in the numerator and denominator.

### Final Answer

After all simplifications, we arrive at:

\[ 8 \cot(6x). \]