\begin{enumerate}
\item $\begin{aligned}[t]
 & \cos^2\dfrac{\pi}{5}-\cos^2\dfrac{4\pi}{5}+\sin^2\dfrac{\pi}{7}-\sin^2\dfrac{8\pi}{7}\\
 =\ & \cos^2\dfrac{\pi}{5}-\cos^2\left(\pi - \dfrac{\pi}{5}\right) +\sin^2\dfrac{\pi}{7}-\sin^2\left(\pi + \dfrac{\pi}{7} \right)\\
 =\ & \cos^2\dfrac{\pi}{5}-\cos^2\dfrac{\pi}{5} +\sin^2\dfrac{\pi}{7}-\sin^2 \dfrac{\pi}{7}\\
 =\ & 0.
 \end{aligned}$
 
 Explications :
 \begin{itemize}
 \item $\cos(\pi-x)=-\cos x$ donc $\big[\cos(\pi-x)\big]^2=\big[-\cos x\big]^2 = \cos^2 x$.
  \item $\sin(\pi+x)=-\sin x$ donc $\big[\sin(\pi+x)\big]^2=\big[-\sin x\big]^2 = \sin^2 x$.
 \end{itemize}
 
\item $\begin{aligned}[t]
&\cos\dfrac{\pi}{3}-\cos\dfrac{2\pi}{3}+\sin\dfrac{\pi}{4}-\sin\dfrac{5\pi}{4}\\
=\ & \cos\dfrac{\pi}{3}-\cos\left(\pi-\dfrac{\pi}{3}\right)+\sin\dfrac{\pi}{4}-\sin\left(\pi+\dfrac{\pi}{4}\right)\\
=\ & \cos\dfrac{\pi}{3}+\cos\dfrac{\pi}{3}+\sin\dfrac{\pi}{4}+\sin\dfrac{\pi}{4}\\
=\ & 2\big[\cos\dfrac{\pi}{3} + \sin\dfrac{\pi}{4} \big]\\
=\ & 2\left(\dfrac{1}{2}+\dfrac{\sqrt2}{2}\right)\\
=\ & 1+\sqrt2.
\end{aligned}$
\end{enumerate}