Need help proving a trigometric identity

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voice
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Need help proving a trigometric identity

Postby voice » Thu Oct 09, 2008 9:26 pm UTC

Hi, I was working through some questions in my textbook when i came across this one which i can't quite prove.

The question is as follows:

Prove that: (secA-tanA)(secA+tanA) / cosecA-cotA = cotA+cosecA

So far I have got as far as: sec^2(A)-tan^2(A)/cosecA-cotA

Could someone please point me in the right direction to go from here? ( I don't want a full proof, just a hint at where to go :D)

Thanks for your time.
Last edited by voice on Thu Oct 09, 2008 10:47 pm UTC, edited 2 times in total.

Buttons
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Re: Trigometric proving problem

Postby Buttons » Thu Oct 09, 2008 9:33 pm UTC

Well, the first thing I notice is sec2A-tan2A. Does that simplify to anything?

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NathanielJ
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Re: Trigometric proving problem

Postby NathanielJ » Thu Oct 09, 2008 9:46 pm UTC

For questions like this it's almost always best to write everything out in terms of cos and sin, IMO. It makes spotting identities so much easier. sec2A - tan2A might look scary, but when written in terms of cos and sin it gets demoted to "standard".
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Sabatini
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Re: Trigometric proving problem

Postby Sabatini » Thu Oct 09, 2008 10:28 pm UTC

Seconded. In general, transferring the problem to the world of sines and cosines is a good problem-solving approach.
Yakk wrote:It is clear you can reduce Well Ordering to the well known Traveling Salesman problem (you determine the order you visit the wells like the salesman does the cities). But TSP is NP -- not possible -- so Well Ordering is wrong.

voice
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Re: Trigometric proving problem

Postby voice » Thu Oct 09, 2008 10:44 pm UTC

Buttons wrote:Well, the first thing I notice is sec2A-tan2A. Does that simplify to anything?


Yes. It simplifies to 1, but I don't see where to go from there. cos^2(x)+sin^2(x)=1 perhaps?

NathanielJ wrote:For questions like this it's almost always best to write everything out in terms of cos and sin, IMO. It makes spotting identities so much easier. sec2A - tan2A might look scary, but when written in terms of cos and sin it gets demoted to "standard".


Sabatini wrote:Seconded. In general, transferring the problem to the world of sines and cosines is a good problem-solving approach.


Thanks for the advice, I'll see what i can do...

ST47
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Re: Need help proving a trigometric identity

Postby ST47 » Thu Oct 09, 2008 11:42 pm UTC

If you haven't got it yet: Do you notice anything interesting about the denominator of your left hand side and your right hand side, that would give you an opportunity to simplify further?

Note that I'm assuming you messed up your parentheses and that the denominator is meant to be (csc-cot), not (csc) with the -cot outside the fraction.

Sabatini
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Re: Need help proving a trigometric identity

Postby Sabatini » Fri Oct 10, 2008 1:45 am UTC

Maybe a conjugate or three?
Yakk wrote:It is clear you can reduce Well Ordering to the well known Traveling Salesman problem (you determine the order you visit the wells like the salesman does the cities). But TSP is NP -- not possible -- so Well Ordering is wrong.

macyran
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Re: Need help proving a trigometric identity

Postby macyran » Fri Oct 10, 2008 8:31 pm UTC

It can be broken down in the following way:
=(((1/cosA)-(sinA/cosA))((1/cosA)+(sinA/cosA)))/((1/sinA)-(cosA/sinA))=((1-sinA)(1+sinA)(sinA))/((cos^2A)(1-cosA))
=((1-sin^2A)(sinA))/((cos^2A)(1-cosA))
=((cos^2A)(sinA))/((cos^2A)(1-cosA))
=(sinA)/(1-cosA)
=((1+cosA)(sinA))/((1+cosA)(1-cosA))
=((sinA+cosAsinA))/((1-cos^2A))
=((sinA+cosAsinA))/((sin^2A))
=cscA+cotA

Sabatini
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Re: Need help proving a trigometric identity

Postby Sabatini » Sat Oct 11, 2008 1:06 am UTC

macyran wrote:It can be broken down in the following way:
=(((1/cosA)-(sinA/cosA))((1/cosA)+(sinA/cosA)))/((1/sinA)-(cosA/sinA))=((1-sinA)(1+sinA)(sinA))/((cos^2A)(1-cosA))
=((1-sin^2A)(sinA))/((cos^2A)(1-cosA))
=((cos^2A)(sinA))/((cos^2A)(1-cosA))
=(sinA)/(1-cosA)
=((1+cosA)(sinA))/((1+cosA)(1-cosA))
=((sinA+cosAsinA))/((1-cos^2A))
=((sinA+cosAsinA))/((sin^2A))
=cscA+cotA


Whoa! Did you just give him a complete solution of a homework problem?
Yakk wrote:It is clear you can reduce Well Ordering to the well known Traveling Salesman problem (you determine the order you visit the wells like the salesman does the cities). But TSP is NP -- not possible -- so Well Ordering is wrong.

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Tez
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Re: Need help proving a trigometric identity

Postby Tez » Sat Oct 11, 2008 5:19 am UTC

Here's a (hopefully) helpful hint.

You were on the rigfht track with what you did so far ....try multiplying out the denominator on the LHS then using the trigonometric identities 1+ cot^2(A)=cosec(A), and sec^(A) = tan@(A)+1. You should get LHS=RHS=1

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Pathway
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Re: Need help proving a trigometric identity

Postby Pathway » Sat Oct 11, 2008 5:05 pm UTC

Perhaps my advice is superfluous, but depending on your level it might be ok to
1) Write everything in terms of sines and cosines
2) Use e^x=cos(x)+i*sin(x) to write everything in terms of exponentials
3) Replace cos(x) by a and sin(x) by b
4) Simplify a fraction! Make sure you remember that i^2=-1
5) Change back into sines and cosines.
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AVIATOR
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Re: Need help proving a trigometric identity

Postby AVIATOR » Mon Oct 20, 2008 12:57 pm UTC

If you'd like another hint:

Spoiler:
Once you've simplified the obvious identities and taken everything in terms of sines and cosines, try multiplying your fraction by the same thing on the top and the bottom (ie one) such that it simplifies something. With the right steps, you can finish it in just a few lines.


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