Числитель
1 - sin^6 a - cos^6 a = (1 - sin^2 a)(1 + sin^2 a + sin^4 a) - cos^2 a*cos^4 a =
= cos^2 a*(1 + sin^2 a + sin^4 a) - cos^2 a*cos^4 a =
= cos^2 a*(1 + sin^2 a + sin^4 a - cos^4 a) =
= cos^2 a*(1 + sin^2 a - (cos^2 a - sin^2 a)*(cos^2 a + sin^2 a)) =
= cos^2 a*(1 + sin^2 a - cos^2 a + sin^2 a) =
= cos^2 a*(1 - cos^2 a + 2sin^2 a) = 3sin^2 a*cos^2 a
Знаменатель
1 - sin^4 a - cos^4 a = (1 - sin^2 a)(1 + sin^2 a) - cos^4 a =
= cos^2 a*(1 + sin^2 a) - cos^4 a = cos^2 a*(1 + sin^2 a - cos^2 a) =
= cos^2 a*(1 - cos^2 a + sin^2 a) = 2sin^2 a*cos^2 a
Делим
(3sin^2 a*cos^2 a) / (2sin^2 a*cos^2 a) = 3/2 = 1,5
cos 3x + 3cos x = 2sin 3x
Выводим формулы тройного аргумента
cos 3x = cos (x + 2x) = cos x*cos 2x - sin x*sin 2x =
= cos x*(2cos^2 x - 1) - sin x*2sin x*cos x = cos x*(2cos^2 x - 1 - 2sin^2 x) =
= cos x*(2cos^2 x - 1 - 2sin^2 x + 2 - 2) = cos x*(4cos^2 x - 3)
Аналогично sin 3x = sin x*(3 - 4sin^2 x)
Подставляем
cos x*(4cos^2 x - 3) + 3cos x = 2sin x*(3 - 4sin^2 x)
cos x*(4cos^2 x - 3 + 3) = 2sin x*(3 - 4sin^2 x + 4 - 4)
cos x*4cos^2 x = 2sin x*(4cos^2 x - 1)
Делим все на 2sin x
2ctg x*cos^2 x = 4cos^2 x - 1
1 = 4cos^2 x - 2ctg x*cos^2 x = 2cos^2 x*(2 - ctg x)
Можно выразить cos^2 x через ctg x
cos^2 x = ctg^2 x / (1 + ctg^2 x)
Подставляем
2ctg^2 x*(2 - ctg x)/(1 + ctg^2 x) = 1
4ctg^2 x - 2ctg^3 x = 1 + ctg^2 x
Замена ctg x = t
2t^3 - 4t^2 + t^2 + 1 = 0
2t^3 - 3t^2 + 1 = 0
2t^3 - 2t^2 - t^2 + t - t + 1 = 0
(t - 1)(2t^2 - t - 1) = 0
(t - 1)(t - 1)(2t + 1) = 0
t1 = ctg x = 1; x1 = pi/4 + pi*k
t2 = -1/2; x2 = -arcctg(1/2) + pi*n