Задача 55. Найдите \({\text{t}}{{\text{g}}^2}\alpha + {\text{ct}}{{\text{g}}^2}\alpha \), если \({\text{tg}}\,\alpha -{\text{ctg}}\,\alpha = 3\)
Решение
\({\rm{tg}}\alpha -{\rm{ctg}}\alpha = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\left( {{\rm{tg}}\alpha -{\rm{ctg}}\alpha } \right)^2} = {3^2}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}\alpha -2{\rm{tg}}\alpha \cdot {\rm{ctg}}\alpha + {\rm{ct}}{{\rm{g}}^2}\alpha = 9\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\)\( \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}\alpha -2 + {\rm{ct}}{{\rm{g}}^2}\alpha = 9\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}\alpha + {\rm{ct}}{{\rm{g}}^2}\alpha = 11.\)
Ответ: 11.