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# Some Trig Help

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I have been learning some trig on my own now through an old text book I found, and because it is so old, it doesnt really give very good explanations of how to prove things. Can someone list the steps you would go through to prove an equation like: csc^6(x) - cot^6(x) = 1 + 3csc^2(x)cot^2(x) or something similar? I am really confused, and im not really sure where to start. Any help would be appreciated. (Btw, this is not a homework problem. I am learning it on my own time.)

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Well I don´t even know what csc is but when dealing with functions like sin/cos/... it´s often usefull to rewrite them as exponential functions with imaginary exponents:

cos(x) + i*sin(x) = exp(ix), sin(x) = 0.5*[exp(ix)-exp(-ix)] ...

dunno how csc and cot are written that way but I´m almost certain there is an expression.

Once you´ve written the trig functions it´s rather easy to verify equations since you only need the rules for working with complex numbers and exponential functions.

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Thank you for that. I forgot completely about ol'' Euler...
I also found some help at mathforum.org, so I should be good to go!

later

Simon

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This equation doesn't look like it needs to be written in terms of complex numbers. I think simple trigonometric identities will solve this one.
BTW: If I'm not mistaken, csc(x) is 1 / sin(x).

[edited by - Kentaro on September 22, 2003 10:02:00 PM]

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Hey, I got bored so I did the problem.

I chose the left side. The basic trigonometric identity that is needed is 1 + (cot(x))^2 = (csc(x))^2

So:

=(csc(x))^2(csc(x))^4 - (cot(x))^2(cot(x))^4

=(1 + (cot(x))^2))(csc(x))^4 - ((csc(x))^2 - 1))(cot(x))^4

=(csc(x))^4 + (csc(x))^4(cot(x))^2 - (cot(x))^4(csc(x))^2 + (cot(x))^4

=(csc(x))^4 + (csc(x)^4(cot(x))^2 - (cot(x))^2((csc(x))^2 - 1)(csc(x))^2 + (cot(x))^4

=(csc(x))^4 + (csc(x)^4(cot(x))^2 - ((cot(x))^2(csc(x)^2 - (cot(x))^2)(csc(x))^2 + (cot(x))^4

=(csc(x))^4 + ((csc(x))^4(cot(x))^2 - ((cot(x))^2(csc(x))^4 - (cot(x))^2)((csc(x))^2) + (cot(x))^4

=(csc(x))^4 + (cot(x))^2(csc(x))^2 + (cot(x))^4

=(csc(x))^2(csc(x))^2 + (cot(x))^2(csc(x))^2 + (cot(x))^2(cot(x))^2

=(1 + (cot(x))^2)(csc(x))^2 + (cot(x))^2(csc(x))^2 + (csc(x))^2 - 1)(cot(x))^2

=(csc(x))^2 - (cot(x))^2 + 3(cot(x))^2(csc(x))^2

=1 + 3(cot(x))^2(csc(x))^2

I did it pretty verbose. If you have questions feel free to ask, or you can confirm it if you have already solved it.

EDIT: BTW, you should discard the comment that was posted by Athiest (I am sure he meant well). A solution with complex numbers is unnecessary.

|PicRepository|PicIndex|

[edited by - nervo on September 23, 2003 1:36:48 PM]

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quote:
Original post by boxdot3
(Btw, this is not a homework problem. I am learning it on my own time.)

That''s fine, but please reread the forum FAQ for our policy on homework and homework-like questions. For questions/posts that look like homework problems (which this one does), the forum FAQ asks you to show your own work even if you have not finished the problem. The FAQ also asks that you not ask for the solution, but hints that you can use to find the solution on your own. Even if this isn''t homework, I''d rather you actually learn how to solve the problem for yourself. Perhaps once you can solve more complex problems without help, then you will be able to make better computer/video games, which is the primary core topic of these forums.

Forum FAQ

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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Ok, I''m sorry about that. I will make sure I post what I''ve tried next time.

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quote:
Original post by boxdot3
Ok, I''m sorry about that. I will make sure I post what I''ve tried next time.

Did you compare what you tried with what I did, or does it not make any sense?

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Yeah, I compared it and it was essentially what I did. Thanks for the help!

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quote:
Original post by boxdot3
Ok, I''m sorry about that. I will make sure I post what I''ve tried next time.

Cool!

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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