Question

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

Answer 1

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.