Найти cos (a) ^8-sin (a) ^8, если cos (a) = 1/3

(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 = - 7/9

sin^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