Question

tg9 - tg27 - tg63 +tg81 ,чему равно
Заранее благодарю

Answer 1

$tg9^{circ} - tg27^{circ} - tg63^{circ} + tg81^{circ} = \\ left[ tg(90^{circ} - x) = ctgx, ctg(90^{circ} - x) = tgx right]\\ = tg9^{circ} -(tg27^{circ} + tg(90^{circ} - 27^{circ})) + tg(90^{circ} - 9^{circ}) =\\ = tg9^{circ} + ctg9^{circ} - (tg27^{circ} + ctg27^{circ}) = \\ left[ tga + ctga = frac{sin a}{cos a} + frac{cos a}{sin a} = frac{1}{cos a sin a} = frac{2}{sin2a} right]\\ = frac{2}{sin18^{circ}} - frac{2}{sin54^{circ}} =$

$= frac{2(sin54^{circ} - sin18^{circ})}{sin18^{circ}sin54^{circ}} = \\ left[ sin a - sin b = 2sinfrac{a - b}{2}cos frac{a + b}{2} right]\\ = frac{2*2sin18^{circ}cos36^{circ}}{sin18^{circ}sin54^{circ}} = frac{4sin18^{circ}cos(90^{circ} - 54^{circ})}{sin18^{circ}sin54^{circ}} =frac{4sin18^{circ}sin54^{circ}}{sin18^{circ}sin54^{circ}} = boxed{4}$