To solve the the question we proceed as follows:
From trigonometric laws
[tex](cos x)^2+(sin x)^2=1[/tex]
[tex](cos x)^2+(sin x)^2=1[/tex]
[tex]sin (x-y)=sin[/tex] [tex]x[/tex] [tex]sin[/tex] [tex]y-sin[/tex] [tex]y[/tex] [tex]cos[/tex] [tex]x[/tex]
[tex]cos (x-y)=cos[/tex] [tex]x[/tex] [tex]cos[/tex] [tex]y+sin[/tex] [tex]x[/tex] [tex]xin[/tex] [tex]y[/tex]
si [tex]x=\frac{8}{17}[/tex]
[tex]cos[/tex] [tex]x=sqrt(1-(sin x)^2)=sqrt(1-64/289)=sqrt(\frac{225}{289} )=\frac{15}{17}[/tex]
[tex]cos[/tex] [tex]y=\frac{3}{5}[/tex]
[tex]sin[/tex] [tex]x= sqrt(1- (cos x)^2)= sqrt(1-\frac{9}{25} )=sqrt(\frac{16}{25} )=\frac{4}{5}[/tex]
thus
[tex]tan (x-y)=[sin (x-y)]/[cos (x-y)][/tex]
=[sin x cos y-sin y cos x]/[cos x cos y+sin x sin y]
plugging in the values we obtain:
[tex][8/17 *3/5-4/5*15/7]/[15/17*3/5+8/17*4/5][/tex]
simplifying
[tex][24/85-60/85]/[45/85+32/85][/tex]
[tex]=-\frac{36}{77}[/tex]
