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1 solution
(x+5)/sin(2θ) =(x+3)/sin(θ) =(x+4)/sin(180-3θ)
(x+5)/(2 sin〖(θ) cos(θ) 〗 )=(x+3)/sin(θ)
cosθ=(x+5)/2(x+3)
sin(180-3θ)=sin(3θ)
sin(3θ)=sin(2θ+θ)=sin(2θ) cos(θ)+cos〖(2θ) sin(θ) 〗
2 sin(θ) cos^2(θ)+=sin(θ) cos(2θ)
〖=sin〗(θ)[cos(2θ)+〖2cos〗^2(θ) ]
〖=sin〗(θ)[2 cos^2θ-1+2 cos^2(θ) ]
〖sin(3θ)=sin〗(θ)[4 cos^2θ-1]
4 cos^2θ-1=(2 cos(θ)-1)(2 cos(θ)+1)
(x+3)/sin(θ) =(x+4)/sin(3θ)
(x+3)/sin(θ) =(x+4)/sin(3θ)
(x+3)/sin(θ) =(x+4)/sin(3θ)
(x+3)/sin(θ) =(x+4)/(sin(θ) [4 cos^2θ-1])
4 cos^2θ-1=(x+4)/(x+3) (2 cos(θ)-1)(2 cos(θ)+1)=(x+4)/(x+3)
[(2) (x+5)/2(x+3) -(x+3)/(x+3)][(2) (x+5)/2(x+3) +(x+3)/(x+3)]
[2/((x+3) )][(2x+8)/((x+3) )]=(x+4)/(x+3)
[(4(x+4))/(x+3)^2 ]=(x+4)/(x+3)
[4/((x+3) )]=1/1
4=x+3 x=1