\begin{enumerate}
\item $\begin{aligned}[t]
\left(\cos\dfrac{\pi}{6}+\sin\dfrac{\pi}{6}\right)^2+\left(\cos\dfrac{\pi}{6}-\sin\dfrac{\pi}{6}\right)^2 & = \left(\dfrac{\sqrt3}{2}+\dfrac{1}{2}\right)^2+\left(\dfrac{\sqrt3}{2}-\dfrac{1}{2}\right)^2\\
& = \dfrac{\big(\sqrt3+1\big)^2+\big(\sqrt3-1\big)^2}{4}\\
& = \dfrac{(3+2\sqrt3+1)+(3-2\sqrt3+1)}{4}\\
& = \dfrac{8}{4}\\
& = 2.
\end{aligned}$

\item $\begin{aligned}[t]
f(x) & = \big(\cos x + \sin x\big)^2 + \big(\cos x - \sin x\big)^2\\
& = (\cos x)^2+2\cos x\sin x + (\sin x)^2 + (\cos x)^2-2\cos x\sin x + (\sin x)^2\\
& = 2(\cos x)^2+2(\sin x)^2\\
& = 2\big(\cos^2 x + \sin^2x\big)\\
& = 2 \text{ car } \cos^2 x + \sin^2x=1.
\end{aligned}$

\underline{Donc $f$ est bien une fonction affine (constante).}
\end{enumerate}