Final answer:
An odd function has no cosine terms in its Fourier expansion because the product of an odd function and an even function (cosine) is odd, and the integral of an odd function over a symmetric interval is zero, which nullifies the cosine coefficients.
Step-by-step explanation:
To prove that an odd function has no cosine terms in its Fourier expansion, we can analyze the integral properties of odd and even functions. A Fourier expansion is a way to represent a function as a series of sines and cosines. Cosine functions are even functions, and in a Fourier series, the coefficient of the cosine term at the n-th harmonic is given by the integral of the function times cos(nx) over one period.
Since odd functions are symmetrical with respect to the origin, meaning f(-x) = -f(x), the integral of an odd function multiplied by an even function (like cosine) over its symmetric interval will always be zero. This is because the product of an odd and an even function is an odd function, and the integral of an odd function over a symmetric interval is zero. Therefore, the cosine coefficients for an odd function must be zero, consequently resulting in no cosine terms in its Fourier expansion.