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.
© John Scholes
jscholes@kalva.demon.co.uk
8 July 2002