How to prove sec(x)tan(x)=0 at x=0 for a horizontal tangent

[Robokapp]

im supposed to prove that

y=Cos(x)

and

y=Sec(x)

have horisontal tangents for x=0

i got the derivatives just fine, i proved the first on e with no problem, but what do i do with second? i get to a point where i must prove that Sec(x)Tan(x)=0 but sec x = 1/cos(x) and i can't work with that...i mean how can i divide 1 by something to get zero? it's an asymptote...so what do i do?

i know how to get derivatives, i don't know how to prove that the second one is having the horisontal tangent.

 

[Fermat]

y = secx
y' = secx.tanx = sinx/cos²x

at x = 0, sinx = 0, cosx = 1

y' = 0/1² = 0
==========

 

[Robokapp]

y = secx
y' = secx.tanx = sinx/cos²x

at x = 0, sinx = 0, cosx = 1

y' = 0/1² = 0
==========

okay...i thought that when you have secx*tanx=0 you have to set them like when you have parenthesis...each one at a time. That was probably a better explanation than most teachers would give. Thank you.

What i was trying to do is prove that either tan or sec is some value and the second one is zero, so zero*value=0 but it doesn't work that way.

can i ask though...why was i wrong? i mean if you do (x)(X+1)=0 you're defenetly not wrong to set either one equl to zero...why wasn't my way working? (To be honest I am more interested on what went wrong than what the answer is...i can always copy that from the book but i don't want to).

 

[Fermat]

As far as I can see, you weren't wrong!

You had,

y' = secx*tanx

at x = 0, secx= 1/cosx = 1/1 = 1, and tanx = 0,

So,

y' = 1*0 = 0

what you have is secx = 1 and the other one, tanx = 0