The period of the function `f(x)=(sin x+sin 2x+sin 4x+sin 5x)/(cos x+cos 2x+cos 4x+cos 5x)` is :

Options

A. `pi/3`
B. `pi/4`
C. `pi`
D. None of these

Answer

To find the period of the function

\[ f(x) = \frac{\sin x + \sin 2x + \sin 4x + \sin 5x}{\cos x + \cos 2x + \cos 4x + \cos 5x}, \]

we will follow these steps:

### Step 1: Rearranging the Function
We can rearrange the numerator and denominator:

\[ f(x) = \frac{\sin x + \sin 5x + \sin 2x + \sin 4x}{\cos x + \cos 5x + \cos 2x + \cos 4x}. \]

**Hint:** Group the sine and cosine terms for easier manipulation.

### Step 2: Applying Sum-to-Product Formulas
Using the sum-to-product identities:

- \(\sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\)
- \(\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\)

We apply these formulas to the numerator and denominator.

**Hint:** Remember to apply the formulas systematically to pairs of sine and cosine terms.

### Step 3: Simplifying the Numerator
For the numerator:

1. Combine \(\sin x + \sin 5x\):
\[ \sin x + \sin 5x = 2 \sin\left(\frac{6x}{2}\right) \cos\left(\frac{-4x}{2}\right) = 2 \sin(3x) \cos(2x). \]

2. Combine \(\sin 2x + \sin 4x\):
\[ \sin 2x + \sin 4x = 2 \sin\left(\frac{6x}{2}\right) \cos\left(\frac{-2x}{2}\right) = 2 \sin(3x) \cos(x). \]

Thus, the numerator becomes:
\[ 2 \sin(3x) \left(\cos(2x) + \cos(x)\right). \]

**Hint:** Keep track of the coefficients and ensure you apply the identities correctly.

### Step 4: Simplifying the Denominator
For the denominator:

1. Combine \(\cos x + \cos 5x\):
\[ \cos x + \cos 5x = 2 \cos\left(\frac{6x}{2}\right) \cos\left(\frac{-4x}{2}\right) = 2 \cos(3x) \cos(2x). \]

2. Combine \(\cos 2x + \cos 4x\):
\[ \cos 2x + \cos 4x = 2 \cos\left(\frac{6x}{2}\right) \cos\left(\frac{-2x}{2}\right) = 2 \cos(3x) \cos(x). \]

Thus, the denominator becomes:
\[ 2 \cos(3x) \left(\cos(2x) + \cos(x)\right). \]

**Hint:** Use the same sum-to-product identities for the cosine terms.

### Step 5: Cancelling Common Terms
Now we can simplify \(f(x)\):

\[ f(x) = \frac{\sin(3x) \left(\cos(2x) + \cos(x)\right)}{\cos(3x) \left(\cos(2x) + \cos(x)\right)}. \]

Assuming \(\cos(2x) + \cos(x) \neq 0\), we can cancel this term:

\[ f(x) = \tan(3x). \]

**Hint:** Ensure that the cancellation is valid by checking the conditions under which the denominator is not zero.

### Step 6: Finding the Period
The period of the function \(f(x) = \tan(3x)\) is given by:

\[ \text{Period} = \frac{\pi}{|3|} = \frac{\pi}{3}. \]

**Hint:** Remember that the period of the tangent function is \(\pi\) divided by the coefficient of \(x\).

### Conclusion
Thus, the period of the function \(f(x)\) is

\[ \boxed{\frac{\pi}{3}}. \]