$\int\limits_0^{\pi /2} {{{\cos }^2}xdx}$

Let $I = \int\limits_0^{\pi /2} {{{\cos }^2}xdx}$ …(i) and $I = \int\limits_0^{\pi /2} {{{\cos }^2}\left( {\cfrac{\pi }{2} - x} \right)dx}$

…(ii)
Adding (i) and (ii),

we have
$2I = \int\limits_0^{\pi /2} {{{\cos }^2}xdx} + \int\limits_0^{\pi /2} {{{\sin }^2}xdx} = \int\limits_0^{\pi /2} {\left( {si{n^2}x + {{\cos }^2}x} \right)dx}$

$= \int\limits_0^{\pi /2} {dx = \left[ x \right]_0^{\pi /2}} = \cfrac{\pi }{2}$ $\Rightarrow$ $I = \cfrac{\pi }{4}$