Question

Докажите тождество: tgα * tgβ + (tgα + tgβ)ctg(α+β)=1

Answer 1

$tga\cdot tg\beta +(tga+tg\beta )\cdot ctg(a+\beta )=\\\\=\frac{sina\cdot sin\beta }{cosa\cdot cos\beta }+(\frac{sina}{cosa}+\frac{sin\beta }{cos\beta })\cdot \frac{cos(a+\beta )}{sin(a+\beta )}=\\\\=\frac{sina\cdot sin\beta }{cosa\cdot cos\beta }+\frac{sina\cdot cos\beta +sin\beta \cdot cosa}{cosa\cdot cos\beta }\cdot \frac{cosa\cdot cos\beta -sina\cdot sin\beta }{sina\cdot cos\beta +sin\beta \cdot cosa}=\\\\=\frac{sina\cdot sin\beta }{cosa\cdot cos\beta }+\frac{cosa\cdot cos\beta -sina\cdot sin\beta }{cosa\cdot cos\beta }=$

$=\frac{sina\cdot sin\beta +cosa\cdot cos\beta -sina\cdot sin\beta }{cosa\cdot cos\beta }=\frac{cosa\cdot cos\beta }{cosa\cdot cos\beta }=1\; ;\\\\1=1\; .$