Question

How do you find the exact value of `Cos[sin^-1(1/3) - arcsin(1/2)]`?

Answer 1

`1/6 + sqrt(2/3)`

Explanation:

First we will consider that `sin^{-1} equiv arcsin`. We will need the identity
`cos(a-b)=cos(a)cos(b)+sin(a)sin(b)` and also that
`cos(arcsin(x))=sqrt(1-x^2)`. With all this in mind
`cos(arcsin(1/3)-arcsin(1/2))=1/6 + sqrt(2/3)`