Implicit Differentiation help please

1. The problem statement, all variables and given/known data
If siny=2sinx and (dy/dx)^2=1+3sec^2(y) show that:
by differentiating 1+3sec^2(y) with respect to x, d^2y/dx^2=3sec^2(y)tan(y)


2. Relevant equations



3. The attempt at a solution

 

sorry i didn't write it, i thought it would be useless but i tried differentiating 1+3sec^2(y) and all i got was 3tany(dy/dx)... if thats write then i dont know how to continue

 

i meant right not write*-

 

i also tried finding dy/dx of siny=2sinx and i got 2cos(X)/cos(y), then i squared that to get (dy/dx)^2 and it matched the given one ie. 1+3sec^2(y) which i also found it to be equal to (cos^2y + 3)/cos^2y.
so:
dy/dx=2cos(X)/cos(y)
(dy/dx)^2=1+3sec^2(y)=(cos^2y + 3)/cos^2y.

 

thank you that really helped. this is what i have managed to do with your help:
d/dx(1+3sec^2(y))= 6sec^3y.siny.dy/dx
d/dx(dy/dx)^2=6sec^3y.siny.dy/dx
i cancelled the dy/dx from the RHS and LHS to get:
d2y/dx2= 6 sec^2y tany

thats a great step forward but the answer needed is 3sec^2y tany... I've probably done a silly mistake somewhere there but i cant find it. Thank you so much for your help and tips :D