[tex]sin\alpha+cos\alpha=\sqrt3\ \ \ |()^2\\\\sin^2\alpha+2sin\alpha cos\alpha+cos^2\alpha=3\\\\1+2sin\alpha cos\alpha=3\\\\2sin\alpha cos\alpha=3-1\\\\2sin\alpha cos\alpha=2\ \ \ |:2\\\\sin\alpha cos\alpha=1[/tex]
[tex]tg\alpha+\frac{1}{tg\alpha}=tg\alpha+ctg\alpha=\frac{sin\alpha}{cos\alpha}+\frac{cos\alpha}{sin\alpha}=\frac{sin^2\alpha+cos^2\alpha}{sin\alpha cos\alpha}=\frac{1}{1}=1[/tex]
