1. It is known that $\operatorname{tg} \alpha+\operatorname{ctg} \alpha=$ and $\alpha \in(0, \pi / 2)$. Find $\frac{1}{\sin \alpha}+\frac{1}{\cos \alpha}$.
Solution. $\frac{\sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha}=a \Rightarrow \frac{1}{\sin \alpha \cos \alpha}=a \Rightarrow \sin 2 \alpha=\frac{2}{a}$. Next,

$\frac{1}{\sin \alpha}+\frac{1}{\cos \alpha}=\frac{\sin \alpha+\cos \alpha}{\sin \alpha \cos \alpha} ;(\sin \alpha+\cos \alpha)^{2}=1+\sin 2 \alpha=\frac{a+2}{a} \Rightarrow \frac{1}{\sin \alpha}+\frac{1}{\cos \alpha}=a \sqrt{\frac{a+2}{a}}=$ $\sqrt{a^{2}+2a}$