sin(x-y) = (24-14*sqrt(2))/75
Write down what you know
sin(x) = 1/3
sec(y) = 25/24
cos(y) = 1/sec(y) = 24/25
cos(x) = sqrt(1-sin(x)^2) = sqrt(1-1/9) = sqrt(8/9) = 2*sqrt(2)/3
sin(y) = sqrt(1-cos(y)^2) = sqrt(1-576/625) = sqrt(49/625) = 7/25
We now know the sin and cos of both x and y.
Now to get the sin of x-y.
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)
Substitute the known values for sin and cos of x and y, then evaluate and simplify
sin(x-y) = (1/3)(24/25) - (2*sqrt(2)/3)(7/25)
sin(x-y) = 24/75 - 14*sqrt(2)/75
sin(x-y) = (24-14*sqrt(2))/75
