Here,
sin7x+sin4x+sinx=0 , where , x in(0,pi/2)sin7x+sin4x+sinx=0,where,x∈(0,π2)
=>sin7x+sinx+sin4x=0⇒sin7x+sinx+sin4x=0
=>2sin((7x+x)/2)cos((7x-x)/2)+sin4x=0⇒2sin(7x+x2)cos(7x−x2)+sin4x=0
=>2sin4xcos3x+sin4x=0⇒2sin4xcos3x+sin4x=0
=>sin4x(2cos3x+1)=0⇒sin4x(2cos3x+1)=0
=>sin4x=0 or2cos3x=-1⇒sin4x=0or2cos3x=−1
=>sin4x=0 orcos3x=-1/2⇒sin4x=0orcos3x=−12
(1)sin4x=0=>2sin2xcos2x=0(1)sin4x=0⇒2sin2xcos2x=0
=>sin2x=0 or cos2x=0⇒sin2x=0orcos2x=0
=>2sinxcosx=0 or 2cos^2x-1=0⇒2sinxcosx=0or2cos2x−1=0
=>sinx=0 or cosx=0 orcos^2x=1/2⇒sinx=0orcosx=0orcos2x=12
=>sinx=0 or cosx=0, cosx=-1/sqrt2 or cosx=1/sqrt2⇒sinx=0orcosx=0,cosx=−1√2orcosx=1√2
color(red)((i)sinx=0=>x=0 !in(0,pi/2)(i)sinx=0⇒x=0∉(0,π2)
color(red)((ii)cosx=0=>x=pi/2!in(0,pi/2)(ii)cosx=0⇒x=π2∉(0,π2)
color(red)((iii)cosx=-1/sqrt2 < 0=>x!in(0,pi/2)(iii)cosx=−1√2<0⇒x∉(0,π2)
color(blue)((iv)cosx=1/sqrt2=>x=pi/4 in(0,pi/2)(iv)cosx=1√2⇒x=π4∈(0,π2)
(2)cos3x=0=>4cos^3x-3cosx=0(2)cos3x=0⇒4cos3x−3cosx=0
=>cosx(4cos^2x-3)=0⇒cosx(4cos2x−3)=0
=>cosx=0 or 4cos^2x=3⇒cosx=0or4cos2x=3
=>cosx=0 or cos^2x=3/4=(sqrt3/2)^2⇒cosx=0orcos2x=34=(√32)2
=>cosx=0 or cosx=-sqrt3/2 or cosx=sqrt3/2⇒cosx=0orcosx=−√32orcosx=√32
color(red)((i)cosx=0=>x=pi/2 !in(0,pi/2)(i)cosx=0⇒x=π2∉(0,π2)
color(red)((ii)cosx=-sqrt3/2 <0=>x !in (0,pi/2)(ii)cosx=−√32<0⇒x∉(0,π2)
color(blue)((iii)cosx=sqrt3/2=>x=pi/6 in(0,pi/2)(iii)cosx=√32⇒x=π6∈(0,π2)
But ,
x=pi/6=>sin7(pi/6)+sin4(pi/6)+sin(pi/6)!=0x=π6⇒sin7(π6)+sin4(π6)+sin(π6)≠0
So, color(red)(x!=pi/6x≠π6
Hence,
x=pi/4 x=π4