Find the fundamental period of the following function:
`f(x)=(sin12x)/(1+cos^(2)6x)`

Explanation

To find the fundamental period of the function \( f(x) = \frac{\sin(12x)}{1 + \cos^2(6x)} \), we will analyze the periods of the numerator and the denominator separately and then use them to find the fundamental period of the entire function.

### Step 1: Find the period of the numerator \( \sin(12x) \)

The standard period of \( \sin(x) \) is \( 2\pi \). For \( \sin(ax) \), the period is given by:

\[ \text{Period} = \frac{2\pi}{|a|} \]

In our case, \( a = 12 \). Therefore, the period of \( \sin(12x) \) is:

\[ \text{Period of } \sin(12x) = \frac{2\pi}{12} = \frac{\pi}{6} \]

### Step 2: Find the period of the denominator \( 1 + \cos^2(6x) \)

The period of \( \cos(x) \) is \( 2\pi \). For \( \cos(ax) \), the period is:

\[ \text{Period} = \frac{2\pi}{|a|} \]

Here, \( a = 6 \), so the period of \( \cos(6x) \) is:

\[ \text{Period of } \cos(6x) = \frac{2\pi}{6} = \frac{\pi}{3} \]

However, we need the period of \( \cos^2(6x) \). The period of \( \cos^2(x) \) is \( \pi \), so the period of \( \cos^2(6x) \) is:

\[ \text{Period of } \cos^2(6x) = \frac{\pi}{3} \]

### Step 3: Determine the fundamental period of the function \( f(x) \)

To find the fundamental period of the function \( f(x) = \frac{\sin(12x)}{1 + \cos^2(6x)} \), we need to find the least common multiple (LCM) of the periods of the numerator and the denominator.

- The period of the numerator \( \sin(12x) \) is \( \frac{\pi}{6} \).
- The period of the denominator \( 1 + \cos^2(6x) \) is \( \frac{\pi}{3} \).

To find the LCM of \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \):

1. Convert both periods to a common denominator:
- \( \frac{\pi}{6} = \frac{\pi}{6} \)
- \( \frac{\pi}{3} = \frac{2\pi}{6} \)

2. The LCM of \( \frac{\pi}{6} \) and \( \frac{2\pi}{6} \) is \( \frac{2\pi}{6} = \frac{\pi}{3} \).

### Conclusion

Thus, the fundamental period of the function \( f(x) = \frac{\sin(12x)}{1 + \cos^2(6x)} \) is:

\[ \text{Fundamental Period} = \frac{\pi}{3} \]

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