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

Question

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

Answer 1

Cos²x+cos²2x=1,  cos²2x=1-cos²x
(cos2x)²=sin²x
(1-2sin²x)²=sin²x
1-4sin²x+4sin⁴x-sin²x=0
4sin⁴x-5sin²x+1=0
sin²x=t, t∈[-1;1]
4t²-5t+1=0
D=25-16=9
t₁=1, t₂=1/4
t=1
sin²x=1, sinx=+-1
sinx=-1, x=-π/2+2πn, n∈Z
sinx=1, x=π/2+2πn, n∈Z
t=1/4
sin²x=1/4, sinx=+-1/2
sinx=-1/2, x=(-1)^(n+1)*π/6+πn, n∈Z
sinx=1/2, x=(-1)^n*π/6+πn, n∈Z
 ответ: x₁=-π/2+2πn, n∈Z
             x₂=π/2+2πn, n∈Z
             x₃=(-1)^(n+1)*π/6+πn, n∈Z
             x₄=(-1)^n*π/6+πn, n∈Z

Answer 2

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