Question 544558
The trigonometric ratios you were given are part of known Pythagorean triples, sets of 3 numbers that can be the lengths of the sides of a right triangle.
{{{drawing(500,200,-5,20,-1,9,
triangle(-4,0,-1,0,-1,4),
triangle(0,0,15,0,15,8),
locate(-2.5,0,3),locate(-0.8,2,4),locate(-3,2.5,5),
locate(7,0,15),locate(7,5,17),locate(15.5,4,8))}}} {{{sin(x)=8/17}}}->{{{cos(x)=15/17}}}{{{tan(x)=8/15}}}, {{{cos(y)=3/5}}}->{{{sin(x)=4/5}}}{{{tan(x)=4/3}}}
Then apply one of those formulas, like
{{{tan(x-y)=(tan(x)-tan(y))/(1+tan(x)tan(y))=(8/15-4/3)/(1+(8/15)(4/3))=(-12/15)/(1+32/45)=(-3/5)/(77/45)=-3*45/(5*77)=-36/77}}}