cos (sin^-1 1/3 - tan^-1 1/2)

For the exact answer use the fact that cos(a-b) = cos(a)cos(b)+sin(a)sin(b)

With a = sin^-1(1/3)  we have cos(a) = (2√2)/3  and sin(a) = 1/3  because if opposite = 1 and hypotenuse = 3 then adjacent = √(3²-1²) = √8 = 2√2

With b = tan^-1(1/2) we have cos(b) = (2√5)/5 and sin(b) = (√5)/5 because if opposite = 1 and adjacent = 2 then hypotenuse = √(1²+2²) = √5  (and 1/√5 = (√5)/5).

I'll leave you to combine the above values to get the final expression. 

cos(asin(1/3) - atan(1/2)) = 0.992345241212.