Which one of the following is not periodic ?

Options

A. `|sin 3x|+sin^(2)x`
B. `cos sqrt(x)+cos^(2)x`
C. `cos 4x+tan^(2)x`
D. `cos 2x + sinx `

Answer

To determine which of the given functions is not periodic, we need to analyze each option based on the definition of periodic functions. A function \( f(x) \) is periodic if there exists a positive number \( T \) such that for all \( x \):

\[ f(x) = f(x + T) \]

### Step-by-Step Solution:

1. **Identify the Functions**: We need to evaluate the periodicity of the following functions:
- \( \sin x \)
- \( \sin^2 x \)
- \( \cos(\sqrt{x}) \)
- \( \cos(4x) \)
- \( \cos(2x) \)

2. **Evaluate \( \sin x \)**:
- The function \( \sin x \) is periodic with a period of \( 2\pi \).
- Therefore, \( \sin x = \sin(x + 2\pi) \).

3. **Evaluate \( \sin^2 x \)**:
- The function \( \sin^2 x \) can be expressed as \( \frac{1 - \cos(2x)}{2} \).
- Since \( \cos(2x) \) is periodic with a period of \( \pi \), \( \sin^2 x \) is also periodic with a period of \( \pi \).

4. **Evaluate \( \cos(\sqrt{x}) \)**:
- The function \( \cos(\sqrt{x}) \) is not periodic. As \( x \) increases, \( \sqrt{x} \) increases without bound, and there is no \( T \) such that \( \cos(\sqrt{x}) = \cos(\sqrt{x + T}) \) for all \( x \).
- Thus, \( \cos(\sqrt{x}) \) does not repeat its values.

5. **Evaluate \( \cos(4x) \)**:
- The function \( \cos(4x) \) is periodic with a period of \( \frac{2\pi}{4} = \frac{\pi}{2} \).
- Therefore, \( \cos(4x) = \cos(4(x + \frac{\pi}{2})) \).

6. **Evaluate \( \cos(2x) \)**:
- The function \( \cos(2x) \) is periodic with a period of \( \frac{2\pi}{2} = \pi \).
- Therefore, \( \cos(2x) = \cos(2(x + \pi)) \).

7. **Conclusion**:
- Among the functions evaluated, \( \cos(\sqrt{x}) \) is the only function that is not periodic.

### Final Answer:
The function that is not periodic is \( \cos(\sqrt{x}) \).