Explanation

(c)
Let $I = \int_{- \pi}^{\pi}\frac{\cos^{2}x}{1 + a^{x}}dx,\ a > 0$ ...(i)
$I = \int_{- \pi}^{\pi}\frac{\cos^{2}x}{1 + a^{- x}}dx$ …(ii)
On adding Eqs. (i)and (ii), we get
$2I = \int_{- \pi}^{\pi}{\cos^{2}{xdx}}$
$= \int_{- \pi}^{\pi}\frac{(\cos{2x + 1)}}{2}dx$
$\Rightarrow \ 2I = \frac{1}{2}\left\lbrack \left( \frac{\sin{2x}}{2} + x \right) \right\rbrack_{- \pi}^{\pi} = \frac{1}{2}(\pi + \pi)$
$\Rightarrow \ I = \frac{\pi}{2}$