سؤال اليوم - 68 -(من فيتنام )

Solve the equation sin⁶x + cos⁶x = 1/4.

السلام عليكم ورحمة الله وبركاته

شكرا لحل الاستاذ شحات حل مضبوط
اليك الحل من الاستاذ Suat Namli

Solve the equation sin⁶x + cos⁶x = 1/4.
Solution
Put a = sin²x, b = cos²x. Then a and b are non-negative with sum 1, so we may put a = 1/2 + h, b = 1/2 - h. Then a³ + b³ = 1/4 + 3h² ≥ 1/4
with equality iff h = 0.
Hence x is a solution of the equation given iff
sin²x = cos²x = 1/2 or x is an odd multiple of π/4.
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