вычислить tg9° - tg 27° - tg63° + tg81°

1. Воспользуемся формулами:

tg (π/2 - a) = ctga; sin (π/2 - a) = cosa; sin (2a) = 2sina * cosa; sina - sinb = 2sin ((a - b) / 2) * cos ((a + b) / 2); x = tg9° - tg27° - tg63° + tg81°; x = tg9° - tg27° - ctg27° + ctg9°; x = sin9°/cos9° - sin27°/cos27° - cos27°/sin27° + cos9°/sin°9.

2. Приведем дроби к общему знаменателю:

x = (sin^2 (9°) + cos^2 (9°)) / (sin9°cos9°) - (sin^2 (27°) + cos^2 (27°)) / (sin27°cos27°); x = 1 / (sin9°cos9°) - 1 / (sin27°cos27°); x = 2/sin18° - 2/sin54°; x = 2 (sin54° - sin18°) / (sin18°sin54°); x = 4sin18°cos36° / (sin18°cos36°); x = 4.

Ответ: 4.