Решить уравнение

tgx-4sin2x-2sin^2x=2cos^2x-ctgx

Tg (x) = sinx/cosx

ctg (x) = cosx/sinx

sin (2x) = 2sinx*cosx

sin^2 (x) + cos^2 (x) = 1

(sinx/cosx) - 8sinx*cosx = 2 (sin^2 (x) + cos^2 (x)) - (cosx/sinx)

(sinx/cosx) - 8sinx*cosx + (cosx/sinx) - 2 = 0

(sin^2 (x) + cos^2 (x)) / (sinx*cosx) - 8sinx*cosx - 2 = 0

(1 - 8 (sinx*cosx) ^2 - 2sinx*cosx) / (sinx*cosx) = 0

1 - 2 * (2sinx*cosx) ^2 - sin (2x) = 0

1 - 2sin^2 (2x) - sin (2x) = 0

Замена: sin (2x) = t, t∈[-1; 1]

-2t^2 - t + 1 = 0

2t^2 + t - 1 = 0, D = 1 + 4*2 = 9

t1 = (-1 + 3) / 4 = 2/4 = 0.5

t2 = (-1 - 3) / 4 = - 4/4 = - 1

1) sin (2x) = 0.5

2x = π/6 + 2πk, x=π/12 + πk

2x = 5π/6 + 2πk, x=5π/12 + πk

2) sin (2x) = - 1

2x = π/2 + 2πk, x=π/4 + πk