Ответы на вопрос:
(cos a)^8 - (sin a)^8 = [(cos a)^4 - (sin a)^4]*[(cos a)^4 + (sin a)^4] == (cos^2 a - sin^2 a)(cos^2 a + sin^2 a)*(cos^4 a + sin^4 a) == (cos 2a)*1*(cos^4 a + 2sin^2 a*cos^2 a + sin^4 a - 2sin^2 a*cos^2 a) == cos 2a *[ (cos^2 a + sin^2 a)^2 - 0,5*4sin^2 a*cos^2 a ] == cos 2a *(1^2 - 1/2*(sin 2a)^2) = cos 2a *(1 - 1/2*sin^2 (2a)) == cos 2a - 1/2*cos 2a*sin^2 (2a)(cos a)^8 - (sin a)^8 = cos 2a - 1/2*cos 2a*sin^2 (2a)теперь подставляем. так как cos a = 1/3, то: cos 2a = 2cos^2 a - 1 = 2*1/9 - 1 = -7/9sin^2 (2a) = 1 - cos^2 (2a) = 1 - 49/81 = 32/81(cos a)^8 - (sin a)^8 = -7/9 - 1/2*(-7/9)*32/81 = -7/9 + 16*7/(9*81) == (-7*81+16*7)/729 = -455/729
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