\cos^{2}{\frac{\pi}{5}}+\cos^{2}{\frac{2\pi}{5}}=
\left(\cos{\frac{\pi}{5}}-\cos{\frac{2\pi}{5}}\right)^2+2\cos\frac{\pi}{5}\cos{\frac{2\pi}{5}}=\\
4\sin^2{\frac{3\pi}{10}}\sin^2{\frac{\pi}{10}}+\frac{2\sin\frac{\pi}{5}\cos\frac{\pi}{5}\cos\frac{2\pi}{5}}{\sin\frac{\pi}{5}}=\\
4\cos^2{\frac{2\pi}{10}}\cos^2{\frac{4\pi}{10}}+\frac{2\sin\frac{2\pi}{5}\cos\frac{2\pi}{5}}{2\sin\frac{\pi}{5}}=
\frac{4\sin^2{\frac{4\pi}{10}}\cos^2{\frac{4\pi}{10}}}{4\sin^2\frac{2\pi}{10}}+\frac{\sin\frac{4\pi}{5}}{2\sin\frac{\pi}{5}}=
\frac{\sin^2{\frac{8\pi}{10}}}{4\sin^2{\frac{2\pi}{10}}}+\frac{2}{4}=
\quad\frac{1}{4}+\frac{2}{4}=\frac{3}{4}