Réponse :
Explications étape par étape
1)
[tex]\sin\alpha+\cos\alpha=-\sqrt2\\(\sin\alpha+\cos\alpha)^2=\sin^2\alpha+2\sin\alpha\cos\alpha+\cos^2\alpha\\(\sin\alpha+\cos\alpha)^2=1+2\sin\alpha\cos\alpha=2\\\\\sin\alpha\cos\alpha=\frac{1}{2}[/tex]
2)
[tex](\sin\alpha-\cos\alpha)^2=\sin^2\alpha-2\sin\alpha\cos\alpha+\cos^2\alpha\\(\sin\alpha-\cos\alpha)^2=1-2\sin\alpha\cos\alpha=1-2\times\frac{1}{2}=0\\\\\Rightarrow\sin\alpha-\cos\alpha=0 \quad\Longleftrightarrow\quad\sin\alpha=\cos\alpha\\\\\sin\alpha+\cos\alpha=-\sqrt2\Rightarrow\left\{\begin{array}{l}\sin\alpha=-\frac{\sqrt2}{2} \\\cos\alpha=-\frac{\sqrt2}{2}\end{array}\right.[/tex]
3)
[tex]\alpha=\pi+\frac{\pi}{4}\\ \\\alpha=\frac{5\pi}{4}[/tex]
