Identify the correct statement the fundamental period of `f(x)=cos(sinx)+cos(cosx)` is `pi`

Options

A. `pi`
B. `2pi`
C. `pi//2`
D. `4pi`

Explanation

To determine the fundamental period of the function \( f(x) = \cos(\sin x) + \cos(\cos x) \), we will follow these steps:

### Step 1: Understand the definition of the fundamental period
The fundamental period \( T \) of a function \( f(x) \) is the smallest positive value such that:
\[ f(x + T) = f(x) \quad \text{for all } x. \]

### Step 2: Check if \( T = \pi \) is a period
We need to evaluate \( f(x + \pi) \):
\[ f(x + \pi) = \cos(\sin(x + \pi)) + \cos(\cos(x + \pi)). \]
Using the properties of sine and cosine:
- \( \sin(x + \pi) = -\sin x \)
- \( \cos(x + \pi) = -\cos x \)

Thus, we have:
\[ f(x + \pi) = \cos(-\sin x) + \cos(-\cos x). \]
Using the even property of cosine, we get:
\[ f(x + \pi) = \cos(\sin x) + \cos(\cos x) = f(x). \]
This shows that \( f(x) \) is periodic with period \( \pi \).

### Step 3: Check if \( T = \pi \) is the least period
Next, we need to check if there is a smaller period, such as \( T = \frac{\pi}{2} \):
\[ f(x + \frac{\pi}{2}) = \cos(\sin(x + \frac{\pi}{2})) + \cos(\cos(x + \frac{\pi}{2})). \]
Using the properties:
- \( \sin(x + \frac{\pi}{2}) = \cos x \)
- \( \cos(x + \frac{\pi}{2}) = -\sin x \)

Thus, we have:
\[ f(x + \frac{\pi}{2}) = \cos(\cos x) + \cos(-\sin x). \]
Using the even property of cosine again:
\[ f(x + \frac{\pi}{2}) = \cos(\cos x) + \cos(\sin x). \]
This is not equal to \( f(x) \), which means \( f(x + \frac{\pi}{2}) \neq f(x) \).

### Conclusion
Since \( f(x + \pi) = f(x) \) and there is no smaller period that satisfies the periodic condition, we conclude that the fundamental period of \( f(x) \) is indeed \( \pi \).

### Final Answer
The statement that the fundamental period of \( f(x) = \cos(\sin x) + \cos(\cos x) \) is \( \pi \) is **correct**.

---