Need help with proving an identity!

**Kiba** · April 24th 2008, 01:13 PM

I need to verify this identity:

(tan x/(1 + cos x)) + (sin x/(1 - cos x)) = cot x + (sec s)(csc x)

I've tried it twice now, and on the left side I keep getting:
sin^2 x /(cot x + (sec x)(csc x)

What I'm doing is multiplying the denominator and the numerator of the first fraction by (1 - cos x) to get sin x^2 on the bottom, and doing the same with the second fraction by multiplying by (1 + cos x). Simplifying that give me the right half of the identity under sin ^2 X

It's supposed to be an identity but I guess I'm screwing it up... can someone try it out and post their work so I can see what I'm doing wrong?

I'd very much appreciate it.

**flyingsquirrel** · April 24th 2008, 01:30 PM

Hi

The idea is the right one :
$\frac{\tan x}{1+\cos x}+\frac{\sin x}{1-\cos x}=\frac{\frac{\sin x}{\cos x}(1-\cos x)+(1+\cos x)\sin x}{\sin^2x}=\frac{\frac{1-\cos x}{\cos x}+1+\cos x}{\sin x}=\ldots$

**Kiba** · April 24th 2008, 01:34 PM

Thanks... but I dont quite understand what you are doing between the second and third step, and I don't know what to do after that....

**Moo** · April 24th 2008, 01:35 PM

Hello,

Divided by sin(x)

**Mathstud28** · April 24th 2008, 01:35 PM

Originally Posted by flyingsquirrel

Hi

The idea is the good one :
$\frac{\tan x}{1+\cos x}+\frac{\sin x}{1-\cos x}=\frac{\frac{\sin x}{\cos x}(1-\cos x)+(1+\cos x)(\sin x)}{\sin^2x}=\frac{\frac{1-\cos x}{\cos x}+1+\cos x}{\sin x}=\ldots$

THe good idea?

Who would have guessed you were French? Bon idee!

**Moo** · April 24th 2008, 01:36 PM

Originally Posted by Mathstud28

THe good idea?

Who would have guessed you were French? Bon idee!

Are you saying that French people can't guess correctly ?

**Mathstud28** · April 24th 2008, 01:37 PM

Originally Posted by Moo

Are you saying that French people can't guess correctly ?

Mais non!

**flyingsquirrel** · April 24th 2008, 01:43 PM

Originally Posted by Kiba

Thanks... but I dont quite understand what you are doing between the second and third step, and I don't know what to do after that....

$\frac{1-\cos x}{\cos x}=\frac{1}{\cos x}-1$ then simplify in the last expression of my first post and you got the result.

Originally Posted by Mathstud28

THe good idea?

Is there another one ?

Bonne idee!

**Mathstud28** · April 24th 2008, 01:44 PM

Originally Posted by flyingsquirrel

$\frac{1-\cos x}{\cos x}=\frac{1}{\cos x}-1$ then simplify in the last expression of my first post and you got the result.

Is there another one ?

Crap...and $\text{bonNE idee}\Rightarrow{right} \text{idea}$
Your sentence would be "This idea is the correct/right one"
Dont worry...today I said ....C'est possible ce(instead of que) je n'ais pas raison"

**Kiba** · April 24th 2008, 01:46 PM

I still don't get it... and I don't know how to simplify that anymore...

**o_O** · April 24th 2008, 03:46 PM

Originally Posted by Mathstud28

Crap...and $\text{bonNE idee}\Rightarrow{right} \text{idea}$
Your sentence would be "This idea is the correct/right one"
Dont worry...today I said ....C'est possible ce(instead of que) je n'aie pas raison"

As for the problem:
$\frac{\frac{\sin x}{\cos x}(1-\cos x)+(1+\cos x)\sin x}{\sin^2x}$

Factor out sin x from the numerator and cancel with the denominator:
$= \frac{\frac{1}{\cos x} - 1 + 1 + \cos x}{\sin x}$

Simplify:
$= \frac{\sec x + \cos x}{\sin x} = \frac{\sec x}{\sin x} + \frac{\cos x}{\sin x}$

I'm sure you can see what happen sfrom here.

**Mathstud28** · April 24th 2008, 04:37 PM

Originally Posted by o_O

As for the problem:
$\frac{\frac{\sin x}{\cos x}(1-\cos x)+(1+\cos x)\sin x}{\sin^2x}$

Factor out sin x from the numerator and cancel with the denominator:
$= \frac{\frac{1}{\cos x} - 1 + 1 + \cos x}{\sin x}$

Simplify:
$= \frac{\sec x + \cos x}{\sin x} = \frac{\sec x}{\sin x} + \frac{\cos x}{\sin x}$

I'm sure you can see what happen sfrom here.

Mon dieu...quest-ce que tu fait? Arrette! Je sais. Je ne suis pas bon a Francais...fiche-moi!

**o_O** · April 24th 2008, 05:01 PM

Originally Posted by Mathstud28

Mon dieu...qu'est-ce que tu fais? Arrête! Je sais. Je ne parle pas bien la français...fiche-moi le camp (?)!

Moo can fix anything else

**Moo** · April 24th 2008, 11:12 PM

He has quite a rude language in French :/ Not only here I mean..
"le français"
or "fiche-moi la paix"

It's perfect

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