Question

Решите доказать ctg(a+b)=(ctg(a)*ctg(b)-1)/(ctg(b)+ctg(a))

Answer 1

$ctg(a+b)=\frac{cos(a+b)}{sin(a+b)}=\frac{cosa \ * \ cosb \ - \ sina \ * \ sinb}{sina \ * \ cosb \ + \ cosa \ * \ sin b}=\\ \\ =\frac{\frac{cosa \ * \ cosb \ - \ sina \ * \ sinb}{sina\ *\ sinb}}{\frac{sina \ * \ cosb \ + \ cosa \ * \ sin b}{sina\ * \ sinb}}=\frac{\frac{cosa \ * \ cosb}{sina\ *\ cosb}-1}{\frac{cosb}{sin b}+\frac{cosa}{sina}}=\frac{ctg4*ctgb-1}{ctgb+ctga}$