Question

Can y'all check my answer?

Answer 1

$\cos x\tan x=\sec x\iff\cos x\dfrac{\sin x}{\cos x}=\sec x\iff\sin x=\sec x$

Multiply both sides by $\cos x$ to get

$\sin x\cos x=1$

Recall the double angle identity for sine to write this as

$\dfrac12\sin2x=1\implies\sin2x=2$

But $-1$ for all $x$, which means there are no $x$ for which $\cos x\tan x=\sec x$ holds true. So all the options are indeed counter-examples.