Айниятро исбот намоед:
\(\frac{\operatorname{tg}{2\alpha}+\operatorname{ctg}{3\beta}}{\operatorname{ctg}{2\alpha}+\operatorname{tg}{3\beta}} = \frac{\operatorname{tg}{2\alpha}}{\operatorname{tg}{3\beta}}\)
\(\frac{\operatorname{tg}{2\alpha}+\operatorname{ctg}{3\beta}}{\operatorname{ctg}{2\alpha}+\operatorname{tg}{3\beta}} = \frac{\frac{\sin{2\alpha}}{\cos{2\alpha}}+\frac{\cos{3\beta}}{\sin{3\beta}}}{\frac{\cos{2\alpha}}{\sin{2\alpha}}+\frac{\sin{3\beta}}{\cos{3\beta}}} =\)
\(=(\frac{\sin{2\alpha}}{\cos{2\alpha}}+\frac{\cos{3\beta}}{\sin{3\beta}}):(\frac{\cos{2\alpha}}{\sin{2\alpha}}+\frac{\sin{3\beta}}{\cos{3\beta}}) =\)
\(=\frac{\sin{2\alpha}\sin{3\beta}+\cos{2\alpha}\cos{3\beta}}{\cos{2\alpha}\sin{3\beta}} : \frac{\sin{2\alpha}\sin{3\beta}+\cos{2\alpha}\cos{3\beta}}{\sin{2\alpha}\cos{3\beta}} =\)
\(=\frac{\sin{2\alpha}\cos{3\beta}}{\cos{2\alpha}\sin{3\beta}} = \frac{\sin{2\alpha}}{\cos{2\alpha}}\cdot\frac{\cos{3\beta}}{\sin{3\beta}} =\)
\(=\operatorname{tg}{2\alpha} : \frac{\sin{3\beta}}{\cos{3\beta}} = \operatorname{tg}{2\alpha} : \operatorname{tg}{3\beta} = \frac{\operatorname{tg}{2\alpha}}{\operatorname{tg}{3\beta}}\)
Яъне:
\(\frac{\operatorname{tg}{2\alpha}+\operatorname{ctg}{3\beta}}{\operatorname{ctg}{2\alpha}+\operatorname{tg}{3\beta}} = \frac{\operatorname{tg}{2\alpha}}{\operatorname{tg}{3\beta}}.\)
Айният исбот шуд.