Is cos 2 x sec x equals 2 cos x - sec x an identity?

Yes, it is.

the basic identity is for a double angle relation:

cos 2x = 2 cosx cos x -1

since sec x =1/cos x if we multiply both sides by sec x we get

cos2xsec x = 2cosxcos x/cos x -1/cos x = 2cos x - sec x