Решите уравнение: sin^2 2x+sin^2 3x=sin^2 4x+sin^2 5x

Question

Решите уравнение: sin^2 2x+sin^2 3x=sin^2 4x+sin^2 5x

Answer 1

1/2(1-cos4x)+1/2(1-cos6x)=1/2(1-cos8x)+1/2(1-cos10x)
1-cos4x+1-cos6x=1-cos8x+1-cos10x
cos10x-cos6x +cos8x-cos4x=0
-2sin8xsin2x-2sin6xsin2x=0
-2sin2x(sin8x+sin6x)=0
-2sin2x*2sin7xcosx=0
sin2x=0⇒2x=πn⇒x=πn/2
sin7x=0⇒7x=πn⇒x=πn/7
cosx=0⇒x=π/2+πn