The correct options are
A sinx2=2sinx22cosx22
D sinx=2tanx21+tan2x2
Option A is an identity as sin2x=2sinxcosx⇒sinx2=2sinx22cosx22
Option B is also Identity
Option C is wrong as logxy is define only when x>0 and y>0 or x<0 and y<0. However, if xy<0, RHS will still be defined, hence not an identity.
Option D is wrong as √1−sin2x=±cosx
Hence, options A and B are correct.