Verified answer

[tex]\displaystyle\bf\\tg\alpha +tg\beta =\frac{Sin\alpha }{Cos\alpha } +\frac{Sin\beta }{Cos\beta } =\frac{Sin\alpha \cdot Cos\beta +Cos\alpha Sin\beta }{Cos\alpha Cos\beta } =\\\\\\=\frac{Sin(\alpha+ \beta )}{Cos\alpha \ Cos\beta }[/tex]