Il suffit de développer :
\[
\begin{aligned}
(\cos x+\sin x)^2+(\cos x-\sin x)^2&= \cos^2 x+2 \cos x \times \sin
x+\sin^2 x\\
&\qquad+\cos^2 x-2 \cos x \times \sin
x+\sin^2 x \\[1ex]
&=2(\cos^2 x+\sin^2 x) \\
&=2 \quad \text{car} \quad \cos^2 x+\sin^2 x=1.
\end{aligned}
\]
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