COS $= SIN ($ - 10 degree)
cosine means "sine of complement"
so, cos x = sin(90-x)
If your equation is to be solved, then you need (I'll use x for $, which is an unusual symbol to be used ...)
cos x = sin(x-10)
sin(90-x) = sin(x-10)
90-x = x-10
2x = 100
x = 50