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# Prove cos2(x)+sin2(x) = 1 Tweet

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1. Re: Prove cos2(x)+sin2(x) = 1
Has this thread been resurrected AGAIN?
2. Re: Prove cos2(x)+sin2(x) = 1
(Original post by Jodin)
Showing my ignorance here once again, isn't the usual way to just make use of the derivative of lnx? (which you determine by the definition of e)

(ln(x+h)-lnx)/h etc.

Then of course y = ln(x) so dy/dx = 1/x = e^-y so dx/dy = e^y and hence
d/dy e^y = e^y ?
This really depends on how you define ln tbh. The most common way to prove the derivative of lnx is 1/x is to use the result we are trying to prove.

If you use the very first definition of e (), then the way of showing it is to use this to demonstrate that . Thence . Obviously this isn't totally rigorous in it's current form, but I find it easier than working from first principles with ln
Last edited by TheMagicMan; 22-01-2012 at 18:26.
3. Re: Prove cos2(x)+sin2(x) = 1
Beaten to it. Just checked the OP.

Spoiler:
Show

Prove the identity .

Define to be the function .

Hence,

Finally, since , it follows that it is also identically zero .
Last edited by gff; 22-01-2012 at 18:42. Reason: It's difficult to come up with something not already mentioned. :D
4. Re: Prove cos2(x)+sin2(x) = 1
I have a proof which revolves around Pythagoras' theorem and right-angled triangle trigonometry. I attached in as a photo... sorry about it not being very clearly layed out, I had to put sin^2 and cos^2 because I could find a notation for power functions!
Attached Thumbnails

Last edited by hippogriff; 13-06-2012 at 18:05.
5. Re: Prove cos2(x)+sin2(x) = 1
Is there a proof without words of this? There is one for a^2+b^2 = c^2 (which is kinda the same thing?).
6. Re: Prove cos2(x)+sin2(x) = 1
(Original post by uncapitalised)
Is there a proof without words of this? There is one for a^2+b^2 = c^2 (which is kinda the same thing?).
Yep.

It needs some words to show what's implied. That's the simplest possible version I think.

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