Question

Can y'all check my answer?

Answer 1

`\cos x\tan x=\sec x\iff\cos x(\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

`\frac12\sin2x=1\implies\sin2x=2`

But
`-1<\sin x<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.