Question

does sin2(x)=2 sinx cosx?

Answer 1

Really???
if by sin2(x) you mean sin^2(x) then
not always
clearly, if x=0, sin^x(x) = sin(2x) = 0
Otherwise, you have
sin^2(x) = 2 sinx cosx
sinx = 2 cosx
tanx = 2
x = arctan(2)

Answer 2

What I'm asking is did you solve sin2[sin^-1(pi/6)] and sin[2sin^-1(pi/6)] in the same way?

Answer 3

no. You asked if they were equal.
I showed you that they are, only in two special cases.

Your followup question is in no way similar.
sin^-1(pi/6) makes no sense at all. I mean, it's possible, since pi/6 < 1 so there is some angle such that sin(x) = pi/6.