Simplifying a log expression with identities

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1. Apr 20, 2016

ProfuselyQuarky

I was supposed to simplify the expression $\ln |\cot {x}|+\ln |\tan {x}\cos {x}|$ and apparently it’s wrong. Where’s the mistake? Is it not simplified enough or . . . ?

$\ln |\cot {x}|+\ln |\tan {x}\cos {x}|$

$=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\frac {\sin {x}}{\cos {x}}\cdot \cos {x}|$

$=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\sin {x}|$

$=\ln |\frac {\cos {x}\sin {x}}{\sin {x}}|$

$=\ln |\cos {x}|$

If it’s really obvious, I’ll be happy with a hint. I thought that this was easy, but apparently I was mistaken?

2. Apr 20, 2016

Math_QED

Looks correct though. Note that Ln0 is undefined, so you might want to specify when the equality is correct.

3. Apr 20, 2016

ProfuselyQuarky

Well, I was given the problem with no specifications and would it really affect the answer? I mean, this expression doesn’t deal with $\ln {0}$.

4. Apr 20, 2016

ProfuselyQuarky

So what is the error?

5. Apr 20, 2016

SteamKing

Staff Emeritus
You sure about that? You mean cos x is never equal to zero?

6. Apr 20, 2016

ProfuselyQuarky

Oh, I never said cos x never is equal to zero. $\cos {\frac {\pi}{2}}=0$ and $\cos {\frac {3\pi}{2}}=0$ and so on.

7. Apr 20, 2016

ProfuselyQuarky

But would that affect the answer? How did I simplify wrong?

8. Apr 20, 2016

SteamKing

Staff Emeritus
So what happens to ln |cos x| when x = (2k+1)π/2 ?

9. Apr 20, 2016

ProfuselyQuarky

We get ln 0 which is undefined

10. Apr 20, 2016

ProfuselyQuarky

Oh, wait, so I'd have to specify that $\ln |\cos {x}|$ is only the answer when $\cos {x}\neq0$.

That's great but there wasn't a single example of any of that in my book which is why I was thinking that my error had something to do with using the identities wrong.

11. Apr 20, 2016

Math_QED

The answer is correct, as long as you specify for which values the expression is undefined.

12. Apr 20, 2016

SammyS

Staff Emeritus
Frankly. I don't think this is the problem. After all, the original expression isn't defined for values of x which make cos(x)=0, either.

However, there are values of x for which the original expression is undefined, but are defined for ln(cos(x)) . Throw those out.

13. Apr 20, 2016

Math_QED

Why do you think your expression is wrong in the first place?

14. Apr 20, 2016

ProfuselyQuarky

It was marked wrong.

15. Apr 20, 2016

SammyS

Staff Emeritus
I'm not sure I was very clear in my previous post.

I don't think the problem is:
that $\ln |\cos {x}|$ is only the answer when $\cos {x}\neq0$.​
Those values of x at which $\cos {x}=0$ are not in the domain of $\ \ln |\cot {x}|+\ln |\tan {x}\cos {x}|\$ in the first place.

I think that the problem is that $\ln |\cos {x}|$ is defined for some values of x for which $\ \ln |\cot {x}|+\ln |\tan {x}\cos {x}|\$ is not defined. I think that you must restrict the domain of the answer to eliminate those.

16. Apr 20, 2016

Math_QED

Even then, it shouldn't be marked wrong since the expression itself is correct.

17. Apr 20, 2016

ehild

SammyS was right. The initial and final expressions are not identical, as their ranges are different. The final one is not defined when cos(x)=0. The initial one is not defined when either sin(x) or cos(x) is zero.