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To determine which function **cannot** be expressed as a Fourier series, we need to consider the properties of Fourier series, which are used to represent periodic functions.

Fourier series can represent a function if the following conditions are met:
1. The function is periodic.
2. The function is integrable over a period.
3. The function has a finite number of discontinuities in any given interval.
4. The function has a finite number of maxima and minima in any given interval.

Let's analyze each option:

- **(a) \( \cos x \)**: This is a periodic function with a period of \(2\pi\). It can be represented as a Fourier series.

- **(b) \( \sin^{-1} x \)**: This is the inverse sine function, which is not periodic. Since Fourier series can only represent periodic functions, this function **cannot** be represented as a Fourier series.

- **(c) \( \sin^2 x \)**: This is a periodic function with a period of \( \pi \) (since \( \sin^2 x = \frac{1}{2}(1 - \cos 2x) \)). It can be represented as a Fourier series.

- **(d) \( \frac{x-2}{5} \)**: This is a linear function, and although it is not periodic, a Fourier series can represent such a function if it is considered over a finite interval and extended periodically.

Thus, the correct answer is **II. (b)** \( \sin^{-1} x \). This function cannot be expressed as a Fourier series because it is not periodic.

Would you like more details or have any other questions?

Here are some related questions you might consider:
1. What are the conditions for a function to be represented by a Fourier series?
2. Can non-periodic functions ever be represented by a Fourier series?
3. How is the Fourier series of a function calculated?
4. What is the difference between a Fourier series and a Fourier transform?
5. Why is the periodicity of a function crucial for its Fourier series representation?

**Tip:** When dealing with Fourier series, always check if the function is periodic and if it satisfies Dirichlet’s conditions for convergence.

Which of the following functions cannot be expressed as a Fourier Series? (a) cos(x), (b) sin^(-1)(x), (c) sin^2(x), (d) (x-2)/5

Explore which functions cannot be expressed as a Fourier series, including discussions on periodicity and Dirichlet’s conditions. Learn why the inverse sine function is not suitable for Fourier representation.

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