Ответ:
[tex]\pm \displaystyle\frac{4}{3}[/tex]
Пошаговое объяснение:
[tex]\[{\mathop{\rm tg}\nolimits} x + {\mathop{\rm tg}\nolimits} y = \displaystyle\frac{{\sin x}}{{\cos x}} + \displaystyle\frac{{\sin y}}{{\cos y}} = \displaystyle\frac{{\sin x\cos y + \cos x\sin y}}{{\cos x\cos y}} =[/tex]
[tex]= \displaystyle\frac{{\sin (x + y)}}{{\cos x\cos y}} = \sin (x + y):\displaystyle\frac{1}{5} = 5\sin (x + y) = 4;[/tex]
[tex]\sin (x + y) = \displaystyle\frac{4}{5}.[/tex]
Используя основное тригонометрическое тождество [tex]{\sin ^2}\alpha + {\cos ^2}\alpha = 1,[/tex] получаем
[tex]{\sin ^2}(x + y) + {\cos ^2}(x + y) = 1;[/tex]
[tex]{\cos ^2}(x + y) = 1 - {\left( {\displaystyle\frac{4}{5}} \right)^2} = 1 - \displaystyle\frac{{16}}{{25}} = \displaystyle\frac{9}{{25}};[/tex]
[tex]\cos (x + y) = \pm \displaystyle\frac{3}{5}.[/tex]
Тогда
[tex]{\mathop{\rm tg}\nolimits} (x + y) = \displaystyle\frac{4}{5}:\left( { \pm \displaystyle\frac{3}{5}} \right) = \pm \displaystyle\frac{4}{3}.[/tex]
