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: Simplifying the Numerator
The numerator is \( \sin 8x \cos x - \sin 6x \cos 3x \). We can use the product-to-sum identities:

\[ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \]

Applying this to both terms:

1. For \( \sin 8x \cos x \):
\[ \sin 8x \cos x = \frac{1}{2} [\sin(9x) + \sin(7x)] \]

2. For \( \sin 6x \cos 3x \):
\[ \sin 6x \cos 3x = \frac{1}{2} [\sin(9x) + \sin(3x)] \]

Now substituting these into the numerator:

\[ \text{Numerator} = \frac{1}{2} [\sin(9x) + \sin(7x)] - \frac{1}{2} [\sin(9x) + \sin(3x)] \]

This simplifies to:

\[ \text{Numerator} = \frac{1}{2} [\sin(7x) - \sin(3x)] \]

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

1. For \( \cos x \cos 2x \):
\[ \cos x \cos 2x = \frac{1}{2} [\cos(3x) + \cos(-x)] = \frac{1}{2} [\cos(3x) + \cos x] \]

2. For \( \sin 3x \sin 4x \):
\[ \sin 3x \sin 4x = \frac{1}{2} [\cos(3x - 4x) - \cos(3x + 4x)] = \frac{1}{2} [\cos(-x) - \cos(7x)] \]

Now substituting these into the denominator:

\[ \text{Denominator} = \frac{1}{2} [\cos(3x) + \cos x] - \frac{1}{2} [\cos(-x) - \cos(7x)] \]

This simplifies to:

\[ \text{Denominator} = \frac{1}{2} [\cos(3x) + \cos x - \cos x + \cos(7x)] = \frac{1}{2} [\cos(3x) + \cos(7x)] \]

### Step 3: Final Expression
Now substituting the simplified numerator and denominator back into the function:

\[ f(x) = \frac{\frac{1}{2} [\sin(7x) - \sin(3x)]}{\frac{1}{2} [\cos(3x) + \cos(7x)]} \]

This simplifies to:

\[ f(x) = \frac{\sin(7x) - \sin(3x)}{\cos(3x) + \cos(7x)} \]

### Step 4: Determining Periodicity
To find the periodicity of \( f(x) \), we need to analyze the components:

- The function \( \sin(7x) \) has a period of \( \frac{2\pi}{7} \).
- The function \( \sin(3x) \) has a period of \( \frac{2\pi}{3} \).
- The function \( \cos(3x) \) has a period of \( \frac{2\pi}{3} \).
- The function \( \cos(7x) \) has a period of \( \frac{2\pi}{7} \).

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

Calculating the LCM:

- The LCM of \( \frac{2\pi}{7} \) and \( \frac{2\pi}{3} \) can be calculated by finding the LCM of the denominators 7 and 3, which is 21. Thus, the LCM of the periods is \( \frac{2\pi \cdot 21}{21} = \frac{2\pi}{1} = 2\pi \).

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