Period of the function `f(x) = sin((pi x)/(2)) cos((pi x)/(2))` is

Answer

To find the period of the function \( f(x) = \sin\left(\frac{\pi x}{2}\right) \cos\left(\frac{\pi x}{2}\right) \), we will follow these steps:

### Step 1: Identify the periods of the individual functions
The function \( f(x) \) is a product of two trigonometric functions: \( \sin\left(\frac{\pi x}{2}\right) \) and \( \cos\left(\frac{\pi x}{2}\right) \).

1. The period of \( \sin(kx) \) is given by \( \frac{2\pi}{k} \).
2. The period of \( \cos(kx) \) is also \( \frac{2\pi}{k} \).

### Step 2: Calculate the period of \( \sin\left(\frac{\pi x}{2}\right) \)
For \( \sin\left(\frac{\pi x}{2}\right) \):
- Here, \( k = \frac{\pi}{2} \).
- Therefore, the period is:
\[ T_1 = \frac{2\pi}{\frac{\pi}{2}} = \frac{2\pi \cdot 2}{\pi} = 4 \]

### Step 3: Calculate the period of \( \cos\left(\frac{\pi x}{2}\right) \)
For \( \cos\left(\frac{\pi x}{2}\right) \):
- Again, \( k = \frac{\pi}{2} \).
- Therefore, the period is:
\[ T_2 = \frac{2\pi}{\frac{\pi}{2}} = \frac{2\pi \cdot 2}{\pi} = 4 \]

### Step 4: Determine the overall period of the function
Since both \( \sin\left(\frac{\pi x}{2}\right) \) and \( \cos\left(\frac{\pi x}{2}\right) \) have the same period of 4, the overall period of the function \( f(x) \) is the least common multiple (LCM) of the individual periods.

- The LCM of \( 4 \) and \( 4 \) is \( 4 \).

### Final Answer
Thus, the period of the function \( f(x) = \sin\left(\frac{\pi x}{2}\right) \cos\left(\frac{\pi x}{2}\right) \) is:
\[ \text{Period} = 4 \]

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