$$\operatorname{tg} \left( {\frac{1}{4}\operatorname{arctg} \frac{4}{3}} \right) = \sqrt 5 - 2.$$ $$\operatorname{tg} \left( {\frac{1}{3}\operatorname{arctg} \frac{{\sqrt 5 }}{7}} \right) = \frac{{3\sqrt 3 - 2\sqrt 5 }}{7}$$ $$\operatorname{tg} \left( {\frac{1}{4}\operatorname{arctg} \frac{{\sqrt 5 }}{2}} \right) = \sqrt 6 - \sqrt 5 $$ $$\operatorname{tg} \left( {\frac{1}{4}\operatorname{arctg} \frac{1}{8}} \right) = \frac{{\sqrt 7 }}{7}$$