jika sin x + cos x = a maka nilai sin^3x + cos^3x adalah

sin x + cos x = a
(sin x + cos x)^2 = a^2
(sin x)^2 + 2 sin x cos x + (cos x)^2 = a^2
(sin x)^2 + (cos x)^2 + 2 sin x cos x = a^2
1 + 2 sin x cos x = a^2
2 sin x cos x = -1 + a^2
sin x cos x = (-1 + a^2)/2

a^3 + b^3 = (a+b)(a^2 - ab + b^2)
(sin x)^3 + (cos x)^3 = (sin x + cos x)(  (sin x)^2 - sin x cos x + (cos x)^2  )
(sin x)^3 + (cos x)^3 = a ( (sin x)^2 + (cos x)^2 - ( -1 + a^2 )/2 )
(sin x)^3 + (cos x)^3 = a ( 1 - (-1 + a^2)/2)
(sin x)^3 + (cos x)^3 = a ( 1+ (1 - a^2)/2 )
(sin x)^3 + (cos x)^3 = a ( 2 + 1 - a^2 )/2
(sin x)^3 + (cos x)^3 = a ( 3 - a^2 ) / 2