This supposed identity is false (missing terms).
Multiply both sides by (cos(x))(cos(2x))(cos(4x)). This leaves sin(8x) on the right hand side.
Left hand side becomes:
8sin(x)cos(x)sin(2x)cos(2x)sin(4x)cos(4x)=
(2sin(x)cos(x))(2sin(2x)cos(2x))(2sin(4x)cos(4x))=
(sin(2x))(sin(4x))(sin(8x)).
The right hand side should have been sec(x)sec(2x)sec(4x)sin(2x)sin(4x)(sin(8x).