trig proof?

Question

proove

ln(cosx) = ln(secx)

Answer 1

ln (cosx)
= ln (1/secx)
= ln (secx)^(-1)
= -ln (secx)
Proved

Answer 2

This is not a correct equation.

Perhaps you mean:

ln(cosx) = -ln(secx) = ln(secx)^-1 = ln(1/secx) = ln(cosx)

Answer 3

It doesnt... sec is 1/cos or (cos)^-1 [not to be confused with arcos or cos^-1]

ln(cos)=ln((cos)^-1)

ln(cos) = -ln(cos)

Which is obviously a condradiction. However, seeing as cos is negative and positive, you should have modulus signs in there since logs of negative numbers dont have real solutions.

Answer 4

ln(cos(x)) = ln(sec(x))

Choose the more complex side; the right hand side.

RHS = ln(sec(x))

Change sec(x) in terms of sine and cosines.

RHS = ln( 1/cos(x) )

Use the log identities to split this log quotient into the difference of logs.

RHS = ln(1) - ln(cos(x))
RHS = 0 - ln(cos(x))
RHS = -ln(cos(x))

Hmm.. they don't seem to be equal. Did you forget a minus sign?

Answer 5

Errrmmmm........... I can't (and neither can anybody else) because it simply isn't true ☺

I'm guessing that you really meant
ln(cos(x))=-ln(sec(x)) which is trivial since
sec(x) = 1/cos(x) = cos(x)^(-1) and
log(z^(-1)) = -ln(z)

HTH

Doug