
\begin{enumerate}
\item Calculons  $\cos^2\left(\dfrac{3\pi}{10}\right)+\cos^2\left(\dfrac{\pi}{5}\right)$.

Remarquons avant tout que:
\[
\frac{3\pi}{10}=\frac{5\pi}{10}-\frac{2\pi}{10}=\frac{\pi}{2}-\frac{\pi}{5}.
\]
Ainsi,
\begin{align*}
\cos^2\left(\dfrac{3\pi}{10}\right)+\cos^2\left(\dfrac{\pi}{5}\right) & = \cos^2\left(\frac{\pi}{2}-\frac{\pi}{5}\right)+\cos^2\left(\frac{\pi}{5}\right)\\
& = \sin^2\left(\frac{\pi}{5}\right) + \cos^2\left(\frac{\pi}{5}\right)\\
& = 1.
\end{align*}
\newpage

\item Calculons $\sin^2\left(\frac{6\pi}{7}\right)-\sin^2\left(\frac{\pi}{7}\right)$.

Remarquons avant tout que:
\[
\frac{6\pi}{7}=\frac{7\pi}{7}-\frac{\pi}{7}=\pi-\frac{\pi}{7}.
\]
Ainsi,
\begin{align*}
\sin^2\left(\frac{6\pi}{7}\right)-\sin^2\left(\frac{\pi}{7}\right) & = \sin^2\left(\pi-\frac{\pi}{7}\right) - \sin^2\left(\frac{\pi}{7}\right)\\
& = \sin^2\left(\frac{\pi}{7}\right) - \sin^2\left(\frac{\pi}{7}\right)\\
& = 0.
\end{align*}

\item Calculons $\sin\left(\dfrac{13\pi}{22}\right)-\cos\left(\dfrac{\pi}{11}\right)$.

Remarquons avant tout que:
\[
\dfrac{13\pi}{22} = \dfrac{11\pi}{22} + \dfrac{2\pi}{22}=\dfrac{\pi}{2}+\dfrac{\pi}{11}.
\]
Ainsi, 
\begin{align*}
\sin\left(\dfrac{13\pi}{22}\right)-\cos\left(\dfrac{\pi}{11}\right) & = \sin\left(\dfrac{\pi}{2}+\dfrac{\pi}{11}\right)-\cos\left(\dfrac{\pi}{11}\right)\\
& = \cos\left(\dfrac{\pi}{11}\right)-\cos\left(\dfrac{\pi}{11}\right)\\
& = 0.
\end{align*}

\item Calculons $1+\sin^2\left(\dfrac{\pi}{13}\right)+\sin^2\left(\dfrac{11\pi}{26}\right)$.

Remarquons avant tout que:
\[
\dfrac{11\pi}{26}=\dfrac{13\pi}{26}-\dfrac{2\pi}{26}=\dfrac{\pi}{2}-\dfrac{\pi}{13}.
\]
Ainsi,
\begin{align*}
1+\sin^2\left(\dfrac{\pi}{13}\right)+\sin^2\left(\dfrac{11\pi}{26}\right) & = 1+\sin^2\left(\dfrac{\pi}{13}\right)+\sin^2\left(\dfrac{\pi}{2}-\dfrac{\pi}{13}\right)\\
& = 1+\sin^2\left(\dfrac{\pi}{13}\right)+\cos^2\left(\dfrac{\pi}{13}\right)\\
& = 1+1\\
& = 2.
\end{align*}
\end{enumerate}
