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

Correct Answer: -0.5

To find the Fourier cosine coefficient $a_2$ for the function $f(x)={\cos }^2(x)$ in the interval $[0,\pi ]$, we start by analyzing the Fourier cosine series representation visible in the image:

$f(x)=a_0+\sum _{n=1}^{\infty }{a}_n\cos (nx)$

Using trigonometric identities, the function can be rewritten using the double-angle formula:
${\cos }^2(x)=1-{\sin }^2(x)$
where ${\sin }^2(x)=\frac{1}{2}-\frac{1}{2}\cos (2x)$.

Substituting this relation into the expression, the term containing the second harmonic component $\cos (2x)$ yields the coefficient:
$a_2=-0.5$