Explanation

(a)
$g(x + \pi) = \int_{0}^{\pi + x}{\cos^{4}{t\ dt}}$
$= \int_{0}^{\pi}{\cos^{4}{t\ dt + \int_{\pi}^{\pi + x}{\cos^{4}tdt}}}$
$I_{1} = I_{2}$
where, $I_{1} = g(\pi)and\ I_{2} = \int_{\pi}^{\pi + x}{\cos^{4}{t\ dt}}$
$\Rightarrow I_{2} = \int_{0}^{x}{\cos^{4}{(y + \pi)dy}}$
$= \int_{0}^{x}{\cos^{4}{y\ dy = g(x)}}$
$\therefore g(\pi + x) = g(x) + g(\pi)$