[tex]\cos x\tan x=\sec x\iff\cos x\dfrac{\sin x}{\cos x}=\sec x\iff\sin x=\sec x[/tex]
Multiply both sides by [tex]\cos x[/tex] to get
[tex]\sin x\cos x=1[/tex]
Recall the double angle identity for sine to write this as
[tex]\dfrac12\sin2x=1\implies\sin2x=2[/tex]
But [tex]-1<\sin x<1[/tex] for all [tex]x[/tex], which means there are no [tex]x[/tex] for which [tex]\cos x\tan x=\sec x[/tex] holds true. So all the options are indeed counter-examples.
