Question 749158
sin[tan^-1(5/12)-sin^-1(-1)] 
let x=the angle whose tan=5/12
tan(x)=5/12
hypotenuse=13 (5-12-13 right triangle)
sin(x)=5/13
cos(x)=12/13
..
let y=the angle whose sin=-1
sin(y)=-1
cos(y)=√(1-sin^2(y))=0
...
sin[tan^-1(5/12)-sin^-1(-1)]=sin(x-y)=sin(x)cos(y)-cos(x)sin(y)=5/13*0-12/13*(-1)=12/13
..
Check: (with calculator)
tan(x)=5/12
x=22.62º
sin(y) =-1
y=270º
x-y=(22.62-270)=-247.38 (in Q3)
reference angle:67.38º
sin(x+y)=sin(67.38º)≈0.923..
exact value=12/13≈0.923..