Answer
$P_{4} = 1 - \frac{\pi^{2}}{2!}x^{2} + \frac{\pi^{4}}{4!}x^{4}$
Work Step by Step
$f(x) = cos (\pi x)$
$f(x) = f(0) + f'(0)x + \frac{f^{''}(0)}{2!}x^{2} + \frac{f^{'''}(0)}{3!}x^{3} + ... + \frac{f^{n}(0)}{n!}x^{n}$
$f(0) = cos (0) = 1$
$f^{'}(0) = -\pi sin(0) = 0$
$f^{''}(0) = -(\pi^{2}) cos(0) = -\pi^{2}$
$f^{'''}(0) = \pi^{3} sin(0) = 0$
$f^{''''}(0) = \pi^{4} cos(0) = \pi^{4}$
Maclaurin polynomial for $n=4$ is
$P_{4} = 1 - \frac{\pi^{2}}{2!}x^{2} + \frac{\pi^{4}}{4!}x^{4}$
