Here's one way to do it.

See the diagram below.

this problem is a lot simpler than it seems. at first it seems like you will need the double angle formula, but in fact you don't. remember that cos^-1(1) = 0

so we want tan(sin^-1(4/5))

let sin^-1(4/5) = x

so we want tanx

now if sin^-1(4/5) = x

=> sinx = 4/5 .............we can find tanx using several identities, or just draw the triangle, see below.

we can find the that the missing side of the triangle is 3 using pythagoras' theorem, or we can simply recognize that this is the famous 3-4-5 triangle.

so tanx is simply opposite/adjacent

=> tan[sin^-1(4/5)+cos^-1 (1)] = tan(sin^-1(4/5)) = tanx = 4/3