Question

Решите уравнение cos12x = cos6 х + sin6 х

Answer 1

Cos12x = Cos²6x - Sin²6x = (Cos6x - Sin6x)(Cos6x + Sin6x)

(Cos6x + Sin6x)(Cos6x - Sin6x - 1)=0

1) Cos6x + Sin6x = 0 ⇔ tg6x = -1

   6x = 3π/4 + πn, n∈Z

   x = π/8 + πn/6, n∈Z

2) Cos6x - Sin6x - 1 = 0

    Cos6x - Sin6x = √2(Cos6x*Cos(π/4) - Sin6x*Sin(π/4))=√2Cos(6x + π/4)

√2Cos(6x + π/4) = 1

Cos(6x + π/4) = √2/2

6x + π/4 = ±π/4 + 2πn, n∈Z

x = πn/3 и x = -π/12 + πn/3, n∈Z