Question

Упростить выражение (sin^4 x+cos^4 x-1)/(sin^6 x+cos^6 x-1)

Answer 1

(sin^4 x + cos^4 x - 1)/(sin^6 x + cos^6 x - 1) = ((sin^2 x + cos^2 x)^2 - 2sin^2 x * cos^2 x - 1)/((sin^2 x + cos^2 x)(sin^4 x - sin^2 x * cos^2 x + cos^4 x) - 1) = (1 - 2sin^2 x * cos^2 x - 1)/((sin^2 x + cos^2 x)^2 - 3sin^2 x * cos^2 x - 1) = (2sin^2 x * cos^2 x)/(3sin^2 x * cos^2 x) = 2/3 ;

Answer 2

1) Упростим выражение sin(x)^4+cos(x)^4-1:
sin(x)^4+cos(x)^4-1 = sin(x)^4+cos(x)^4+2sin(x)^2*cos(x)^2-2sin(x)^2*cos(x)^2-1 = (sin(x)^2+cos(x)^2)^2-2sin(x)^2*cos(x)^2-1 = 1-2sin(x)^2*cos(x)^2-1 = -2sin(x)^2*cos(x)^2
2) Упростим sin^6 x+cos^6 x-1:
sin^6 x+cos^6 x-1 = (sin(x)^2+cos(x)^2)(sin(x)^4-sin(x)^2*cos(x)^2+cos(x)^4)-1 = sin(x)^4-sin(x)^2*cos(x)^2+cos(x)^4-1 = (sin(x)^4+cos(x)^4-1)-sin(x)^2*cos(x)^2. Используем упрощенное выражение из пункта 1:
(sin(x)^4+cos(x)^4-1)-sin(x)^2*cos(x)^2 = -2sin(x)^2*cos(x)^2-sin(x)^2*cos(x)^2 = -3sin(x)^2*cos(x)^2.
Сократим дробь:
(-2sin(x)^2*cos(x)^2)/(-3sin(x)^2*cos(x)^2)=2/3.