The function f(x) given by `f(x)=(sin 8x cos x-sin6x cos 3x)/(cos x cos2x-sin3x sin 4x)` , is

Options

A. periodic with period `pi`
B. periodic with period `2 pi`
C. periodic with period `pi//2`
D. not periodic

Answer

To determine the periodicity of the function \( f(x) = \frac{\sin(8x) \cos(x) - \sin(6x) \cos(3x)}{\cos(x) \cos(2x) - \sin(3x) \sin(4x)} \), we will simplify the expression step by step.

### Step 1: Simplify the numerator
The numerator is \( \sin(8x) \cos(x) - \sin(6x) \cos(3x) \). We can use the product-to-sum identities to simplify this.

Using the identity:
\[ \sin A \cos B = \frac{1}{2} (\sin(A+B) + \sin(A-B)) \]
we can rewrite each term:
1. For \( \sin(8x) \cos(x) \):
\[ \sin(8x) \cos(x) = \frac{1}{2} (\sin(9x) + \sin(7x)) \]
2. For \( \sin(6x) \cos(3x) \):
\[ \sin(6x) \cos(3x) = \frac{1}{2} (\sin(9x) + \sin(3x)) \]

Thus, the numerator becomes:
\[ \frac{1}{2} (\sin(9x) + \sin(7x)) - \frac{1}{2} (\sin(9x) + \sin(3x)) = \frac{1}{2} (\sin(7x) - \sin(3x)) \]

### Step 2: Simplify the denominator
The denominator is \( \cos(x) \cos(2x) - \sin(3x) \sin(4x) \). We can also use the product-to-sum identities here.

Using the identity:
\[ \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) \]
we rewrite:
1. For \( \cos(x) \cos(2x) \):
\[ \cos(x) \cos(2x) = \frac{1}{2} (\cos(3x) + \cos(x)) \]
2. For \( \sin(3x) \sin(4x) \):
\[ \sin(3x) \sin(4x) = \frac{1}{2} (\cos(1x) - \cos(7x)) \]

Thus, the denominator becomes:
\[ \frac{1}{2} (\cos(3x) + \cos(x)) - \frac{1}{2} (\cos(1x) - \cos(7x)) = \frac{1}{2} (\cos(3x) + \cos(x) - \cos(1x) + \cos(7x)) \]

### Step 3: Combine the results
Now we can rewrite the function:
\[ f(x) = \frac{\frac{1}{2} (\sin(7x) - \sin(3x))}{\frac{1}{2} (\cos(3x) + \cos(x) - \cos(1x) + \cos(7x))} \]
This simplifies to:
\[ f(x) = \frac{\sin(7x) - \sin(3x)}{\cos(3x) + \cos(x) - \cos(1x) + \cos(7x)} \]

### Step 4: Determine periodicity
To find the period of \( f(x) \), we need to consider the periodicity of the sine and cosine functions involved. The sine and cosine functions have a fundamental period of \( 2\pi \).

1. The terms \( \sin(7x) \) and \( \sin(3x) \) have periods \( \frac{2\pi}{7} \) and \( \frac{2\pi}{3} \) respectively.
2. The terms \( \cos(3x) \), \( \cos(x) \), \( \cos(1x) \), and \( \cos(7x) \) have periods \( \frac{2\pi}{3} \), \( 2\pi \), \( 2\pi \), and \( \frac{2\pi}{7} \) respectively.

The least common multiple (LCM) of the periods will give us the period of the function \( f(x) \).

The LCM of \( \frac{2\pi}{7} \) and \( \frac{2\pi}{3} \) is \( \frac{2\pi}{21} \).

### Conclusion
Thus, the function \( f(x) \) is periodic with a period of \( \frac{\pi}{2} \).