[tex]sin\alpha+cos\alpha=\frac{\sqrt5}{2}\ \ \ |()^2\\\\sin^2\alpha+2sin\alpha cos\alpha+cos^2\alpha=\frac{5}{4}\\\\1+2sin\alpha cos\alpha=\frac{5}{4}\\\\2sin\alpha cos\alpha=\frac{5}{4}-1\\\\2sin\alpha cos\alpha=\frac{1}{4}\ \ \ |:2\\\\sin\alpha cos\alpha=\frac{1}{8}[/tex]
[tex]tg\alpha+\frac{1}{tg\alpha}=\frac{sin\alpha}{cos\alpha}+\frac{1}{\frac{sin\alpha}{cos\alpha}}=\frac{sin\alpha}{cos\alpha}+\frac{cos\alpha}{sin\alpha}=\frac{sin^2\alpha+cos^2}{sin\alpha cos\alpha}=\frac{1}{sin\alpha cos\alpha}=\frac{1}{\frac{1}{8}}=8[/tex]
