Cos^2(x) + cosx = sin^2(x)
cos^2(x) + cosx - (1 - cos^2(x)) = 0
cos^2(x) + cos^2(x) + cosx - 1 = 0
2cos^2(x) + cosx - 1 = 0
замена: cosx = t∈[-1;1]
2t^2 + t - 1 = 0, D=9
t1 = (-1-3)/4 = -1
t2 = (-1+3)/4 = 1/2
1) cosx = -1
x = π + 2πk, k∈Z
2) cosx = 1/2
x = +- π/3 + 2πk, k∈Z
Найдем наибольший отрицательный корень:
k = -1, x = π - 2π = -π; x = π/3 - 2π = -5π/3; x = -π/3 - 2π = -7π/3
k = 0, x = -π/3
-π/3 > -π
Ответ: x = -π/3 - наибольший отрицательный корень