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znanija.com/task/35217707

sin(5x) - sin(x) =  sin(7x) - sin(3x) ⇔2sin(2x)*cos(3x) = 2sin(2x)*cos(5x) ⇔

2sin(2x) *( cos(3x)  - cos(5x) ) =0 ⇔2sin(2x) *2sin(x)*sin(4x)  = 0  ⇔

4sin(4x)*sin(2x) *sin(x)  = 0

sin4x = 0 ⇒ 4x = πn  ⇔    x =(π/4)n , n ∈ ℤ

* * * * *  P.S.

sinα - sinβ =2sin((α -β)/2) *cos((α +β)/2)

sin(2α =2sinα*cosα   ;  sin4α =2sin2α*cos2α= 4sinα*cosα*cos2α