
\begin{align*}
\left(\cos\dfrac{\pi}{5}+\sin\dfrac{\pi}{5}\right)^2+\left(\cos\dfrac{\pi}{5}-\sin\dfrac{\pi}{5}\right)^2
& = \cos^2\dfrac{\pi}{5}+2\cos\dfrac{\pi}{5}\sin\dfrac{\pi}{5} + \sin^2\dfrac{\pi}{5}\\
& \hspace*{4cm} + \cos^2\dfrac{\pi}{5}-2\cos\dfrac{\pi}{5}\sin\dfrac{\pi}{5} + \sin^2\dfrac{\pi}{5}\\
& = 2\underbrace{\left(\cos^2\dfrac{\pi}{5} + \sin^2\dfrac{\pi}{5}\right)}_{=1\text{ car }\cos^2x+\sin^2x=1}\\
& = 2.
\end{align*}
