Do \(\cos \frac{{7\pi }}{8} = \cos \left( {\pi - \frac{\pi }{8}} \right) = - \cos \left( { - \frac{\pi }{8}} \right) = - \cos \frac{\pi }{8}\);

\(\cos \frac{{5\pi }}{8} = \cos \left( {\pi - \frac{{3\pi }}{8}} \right) = - \cos \left( { - \frac{{3\pi }}{8}} \right) = - \cos \frac{{3\pi }}{8}\).

Nên A = \({\cos ^2}\frac{\pi }{8} + {\cos ^2}\frac{{3\pi }}{8} + {\cos ^2}\frac{{5\pi }}{8} + {\cos ^2}\frac{{7\pi }}{8}\)

= \({\cos ^2}\frac{\pi }{8} + {\cos ^2}\frac{{3\pi }}{8} + {\left( { - \cos \frac{{3\pi }}{8}} \right)^2} + {\left( { - \cos \frac{\pi }{8}} \right)^2}\)

\( = 2\left( {{{\cos }^2}\frac{\pi }{8} + {{\cos }^2}\frac{{3\pi }}{8}} \right)\)

\( = 2\left[ {{{\cos }^2}\frac{\pi }{8} + {{\sin }^2}\left( {\frac{\pi }{2} - \frac{{3\pi }}{8}} \right)} \right]\)

\( = 2\left( {{{\cos }^2}\frac{\pi }{8} + {{\sin }^2}\frac{\pi }{8}} \right) = 2.1 = 2\).