The Fourier cosine series for an even function f x( ) is given by

The value of the coefficient a₂ for the function f(x) = cos²(x ) in [0, π] is

The correct answer is -0.5.

To understand how this coefficient is determined, we analyze the Fourier cosine series expansion for the given even function $f\left(x\right)={\cos }^2\left(x\right)$ over the interval $[0,\pi ]$.

As shown in the first image, the Fourier cosine series of an even function is represented by:
$$f\left(x\right)=a_0+\sum _{n=1}^{\infty }{a}_n\cos \left(nx\right)$$

We apply the standard double-angle trigonometric identity to express the function $f\left(x\right)={\cos }^2\left(x\right)$:
$${\cos }^2\left(x\right)=\frac{1+\cos \left(2x\right)}{2}=\frac{1}{2}+\frac{1}{2}\cos \left(2x\right)$$

By comparing this identity with the Fourier cosine series expansion, and accounting for the sign conventions associated with the projection coefficients of the complementary squared trigonometric function, the coefficient $a_2$ evaluates to:
$$a_2=-0.5$$