Tg15°tg30°tg45°tg60°tg75°

Решите задачу:

$tg15^{\circ}tg30^{\circ}tg45^{\circ}tg60^{\circ}tg75^{\circ}= \\ =tg(45^{\circ}-30^{\circ})tg30^{\circ}tg45^{\circ}tg60^{\circ}tg(30^{\circ}+45^{\circ})= \\ = \dfrac{tg45^{\circ}-tg30^{\circ}}{1+tg45^{\circ}tg30^{\circ}}tg30^{\circ}tg45^{\circ}tg60^{\circ} \dfrac{tg30^{\circ}+tg45^{\circ}}{1-tg30^{\circ}tg45^{\circ}} = \\ = \dfrac{tg^245^{\circ}-tg^230^{\circ}}{1-tg^245^{\circ}tg^230^{\circ}} tg30^{\circ}tg45^{\circ}tg60^{\circ} =$
$= \dfrac{1-( \dfrac{ \sqrt{3} }{3})^2 }{1-( \dfrac{ \sqrt{3} }{3})^2 } \cdot \dfrac{ \sqrt{3} }{3}\cdot \sqrt{3} \cdot 1 = \\ =\boxed1$