Solve this equation/[/sin^6 x+/cos^6 x=/frac{5}{8}/]for / (x / in[0,2/pi]/)

Sin^6 x + cos^6 x =

= (sin² x + cos² x) (sin^4 x - sin² x cos² x + cos^4 x) =

= sin^4 x + 2 sin²x cos² x + cos^4 x - 2 sin² x cos² x - sin² x cos² x =

= (sin² x + cos² x) ² - 3 sin² x cos² x =

= 1 - 3 sin² x cos² x

After that:

1 - 3 sin² x cos² x = 5/8

1 - 5/8 = 3 sin² x cos² x

3/8 = 3 sin² x cos² x / : 3

1/8 = sin² x cos² x / * 4

1/2 = 4 sin² x cos² x

1/2 = (2 sin x cos x) ²

√2 / 2 = 2 sin x cos x

sin 2 x = √ 2 / 2

2 x = π / 4, or x = 3 π / 4

Answer:

x 1 = π / 8, x 2 = 3 π / 8.