Ctg2x + Ctg5x = Ctg3x + Ctg4x
(есть формула:Ctgα + Ctgβ = Sin(α+β)/SinαSinβ. Применим)
Sin7x/Sin2xSin5x = Sin7x/Sin3xSin4x
Sin7x/Sin2xSin5x - Sin7x/Sin3xSin4x = 0
Sin7x(1/Sin2xSin5x - 1/Sin3xSin4x) = 0
Sin7x = 0 или (1/Sin2xSin5x - 1/Sin3xSin4x) = 0
7х = πn , n ∈Z (Sin3xSin4x - Sin2xSin5x)/Sin2xSin5xSin3xSin4x =0
x =πn/7, n ∈Z 1)Sin3xSin4x - Sin2xSin5x=0
Sin2xSin5xSin3xSin4x ≠0
2) 1/2(Сosx -Cos7x) -1/2(Cos3x - Cos7x) = 0
Sin2xSin5xSin3xSin4x ≠0
3) (Сosx -Cos7x) - (Cos3x - Cos7x) = 0
Sin2xSin5xSin3xSin4x ≠0
4) Cosx - Cos3x = 0
Sin2xSin5xSin3xSin4x ≠0
5) -2Sin2xSinx = 0
Sin2xSin5xSin3xSin4x ≠0 , ⇒ Sin2x≠0, Sin5x≠0, Sin3x≠0, Sin4x≠0
x = πk , k ∈Z