Establishing an identity: tan(x)/(tan^2(x)-1)=1/(tan(x)-cot(

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Establishing an identity: tan(x)/(tan^2(x)-1)=1/(tan(x)-cot(

Establish the following identity:

$\frac{\tan(x)}{\tan^2(x)\, -\, 1}\, =\, \frac{1}{\tan(x)\, -\, \cot(x)}$

My attempt was(using "Left Hand Side"):
· Convert the tan and tan^2 to sin(x)/cos(x) and sin^2(x)/cos^2(x) respectively.

$\mbox{Step 1. }\, \frac{\frac{\sin(x)}{\cos(x)}}{\frac{\sin^2(x)}{\cos^2(x)}\, -\, 1}$

$\mbox{Step 2. }\, \left(\frac{\sin(x)}{\cos(x)}\right)\, \times \,\left(\frac{\cos^2(x)}{sin^2(x)}\, -\, 1\right)$

(cross multiply flipped denominator?)
Step 3. FOIL the sinx/cosx
Step 4. Fid a common denominator to get:

$\frac{\cos^2(x)\, -\, \sin^2(x)}{\sin(x) \cos(x)}$

I don't see how to simplify any further to get it to match the "Right Hand Side".
tampster

Posts: 3
Joined: Sun Feb 26, 2012 4:35 am

tampster wrote:$\mbox{Step 1. }\, \frac{\frac{\sin(x)}{\cos(x)}}{\frac{\sin^2(x)}{\cos^2(x)}\, -\, 1}$

$\mbox{Step 2. }\, \left(\frac{\sin(x)}{\cos(x)}\right)\, \times \,\left(\frac{\cos^2(x)}{sin^2(x)}\, -\, 1\right)$

(cross multiply flipped denominator?)

Since you did not have the two original terms combined using a common denominator, you cannot flip "the" fraction underneath. Try doing the conversion first, and see where this leads.

. . . . .$\frac{\frac{\sin(x)}{\cos(x)}}{\frac{\sin^2(x)}{\cos^2(x)}\, -\, 1} \, =\, \frac{\left(\frac{\sin(x)}{\cos(x)}\right)}{\left(\frac{\sin^2(x)}{\cos^2(x)}\, -\, \frac{\cos^2(x)}{\cos^2(x)}\right)}$

...and so forth.

stapel_eliz

Posts: 1729
Joined: Mon Dec 08, 2008 4:22 pm

Re: Establishing an identity: tan(x)/(tan^2(x)-1)=1/(tan(x)-

Well, I attempted the problem using your suggestion and I was able to reduce it to this, but do not see any further simplification possibilities:

The denominator could be changed to -cos(2x), but I don't see that helping.
The denominator could be factored to get (sinx-cosx)(sinx+cosx), but I don't see that helping either.

PLZ HELP my HW is due tomorrow
tampster

Posts: 3
Joined: Sun Feb 26, 2012 4:35 am

Re: Establishing an identity: tan(x)/(tan^2(x)-1)=1/(tan(x)-

Try working on the other side. Change this to sines and cosines and do the common denominator. Then turn the dividing into multiplying, and compare what you get with what you already have on the left-hand side. You can then do the proof by showing the steps from where you are on the left-hand side, going backwards through the steps on the right-hand side, until you get to the starting place on the right-hand side.

maggiemagnet

Posts: 305
Joined: Mon Dec 08, 2008 12:32 am

SOLVED Establishing an identity: tan(x)/(tan^2(x)-1)=1/(tan(

Solved:
(original left hand side equation all divided by tan)
=

tampster

Posts: 3
Joined: Sun Feb 26, 2012 4:35 am