Here,
sin7x+sin4x+sinx=0 , where , x in(0,pi/2)
=>sin7x+sinx+sin4x=0
=>2sin((7x+x)/2)cos((7x-x)/2)+sin4x=0
=>2sin4xcos3x+sin4x=0
=>sin4x(2cos3x+1)=0
=>sin4x=0 or2cos3x=-1
=>sin4x=0 orcos3x=-1/2
(1)sin4x=0=>2sin2xcos2x=0
=>sin2x=0 or cos2x=0
=>2sinxcosx=0 or 2cos^2x-1=0
=>sinx=0 or cosx=0 orcos^2x=1/2
=>sinx=0 or cosx=0, cosx=-1/sqrt2 or cosx=1/sqrt2
color(red)((i)sinx=0=>x=0 !in(0,pi/2)
color(red)((ii)cosx=0=>x=pi/2!in(0,pi/2)
color(red)((iii)cosx=-1/sqrt2 < 0=>x!in(0,pi/2)
color(blue)((iv)cosx=1/sqrt2=>x=pi/4 in(0,pi/2)
(2)cos3x=0=>4cos^3x-3cosx=0
=>cosx(4cos^2x-3)=0
=>cosx=0 or 4cos^2x=3
=>cosx=0 or cos^2x=3/4=(sqrt3/2)^2
=>cosx=0 or cosx=-sqrt3/2 or cosx=sqrt3/2
color(red)((i)cosx=0=>x=pi/2 !in(0,pi/2)
color(red)((ii)cosx=-sqrt3/2 <0=>x !in (0,pi/2)
color(blue)((iii)cosx=sqrt3/2=>x=pi/6 in(0,pi/2)
But ,
x=pi/6=>sin7(pi/6)+sin4(pi/6)+sin(pi/6)!=0
So, color(red)(x!=pi/6
Hence,
x=pi/4