Question

d/dx∫ ∫tan(t) dt. Cos (x) is on the top of ∫ and sin(x) is on the bottom of ∫

Answer 1

If F(x) = ∫[u,v] f(t) dt

where u and v are functions of x, then
F'(x) = f(v)*v'(t) - f(u)*u'(t)
This is just the chain rule in disguise. So, for your function
F'(x) = tan(cos x)*(-sin x) - tan(sin x)*(cos x)