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

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1. Re: Prove cos2(x)+sin2(x) = 1
I think for aspiring mathmos its a good idea to try to understand why the proof involving power series is important - its because sin x and cos x are best defined by their power series and therefore the proof is directly from the definition.

Even for A-Level students the proof has value IMO. You might not know the Cauchy product formula but if you accept that you can follow the proof
2. Re: Prove cos2(x)+sin2(x) = 1
IMO the Original poster was just some weird math nerd trolling TSR, I doubt he still posts here.
3. Re: Prove cos2(x)+sin2(x) = 1
IMO the Original poster was just some weird math nerd trolling TSR, I doubt he still posts here.
Yeah, what a douche bag. Good riddance tbh.
4. Re: Prove cos2(x)+sin2(x) = 1
(Original post by nuodai)
Tony Hawken, why are you on TSR? You're not a student, which implies that your goal is to help students; but so far most of your posts seem to be made with the sole purpose of criticising others' responses to threads. This is sometimes okay when the posts you're criticising genuinely are misleading, but in the majority of cases it seems like you're just here for an argument -- they seem to be based on the idea that people only use TSR for help with GCSE/A-level maths problems and that they no-one comes here to just discuss maths, at whatever level necessary, as is the case in this thread. At best it's irritating, and at worst it's going to further confuse people asking questions (especially when you're mistaken, like in the big-O notation thread, or when you've completely missed the point of the thread, like here). In any case it's getting really tiresome.
5. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
You are now changing the subject. If you are required to prove the originally posted trig identity you only need to think of sine and cosine as simple ratios. No one asked about finding the derivative.
Thinking in terms of angles starts to break down when x is complex - the power series proofs do not.
6. Re: Prove cos2(x)+sin2(x) = 1
(Original post by RichE)
Yes, it has been a long time since I have been a student in the real sense, but I think you will find that there are many people like myself posting on these forums. Generally the tone of my postings are quite positive. But, I am allowed to criticise and this I feel I need to when I violently disagree with what someone else has posted - particularly if the tone of their posting is negative.

I don't feel you have a right to dictate who can and can't post on these forums. Neither do you have a right to curb free speech. As long as I obey the rules for this site and behave myself, I feel that I have the right to participate.

Further more, how am I confusing people by insisting that the simplest proofs are used? No, it is the people who resort to much more advanced mathematics to prove what is essentially a very simple trig identity who are likely to confuse.
Last edited by tonyhawken; 29-08-2011 at 15:18.
7. Re: Prove cos2(x)+sin2(x) = 1
(Original post by Glutamic Acid)
Thinking in terms of angles starts to break down when x is complex - the power series proofs do not.
Yes, this may be the case. Then the only time that you are likely to be asked to prove this trig identity is for a C1 paper. They don't currently teach complex numbers until you do further maths.
8. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
Yes, it has been a long time since I have been a student in the real sense, but I think you will find that there are many people like myself posting on these forums. Generally the tone of my postings are quite positive. But, I am allowed to criticise and this I feel I need to when I violently disagree with what someone else has posted - particularly if the tone of their posting is negative.

I don't feel you have a right to dictate who can and can't post on these forums. Neither do you have a right to curb free speech. As long as I obey the rules for this site and behave myself, I feel that I have the right to participate.
I don't deny that, I was just asserting that if you've come here with the view of helping people with maths problems and engaging in conversation about maths, your intentions aren't yet being fulfilled. In the same flavour, I have every right to express my opinion that your posts have been on the whole counterproductive so far (my hopes being that some time in the near future this might change).
Last edited by nuodai; 29-08-2011 at 15:17.
9. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
Yes, this may be the case. Then the only time that you are likely to be asked to prove this trig identity is for a C1 paper. They don't currently teach complex numbers until you do further maths.
I think this would be a reasonable question for a first-year undergraduate analysis course. In this case, attempting a proof by referring to right-angled triangles isn't going to be what's required.
10. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
Yes, this may be the case. Then the only time that you are likely to be asked to prove this trig identity is for a C1 paper. They don't currently teach complex numbers until you do further maths.
There are other scenarios where a proof could be asked such as... this thread? Also, this appeared in a mock IA tripos paper last year: the functions were suggestively introduced as c(z) and s(z) with power series definitions and the question culminated in showing that c(z)^2 + s(z)^2 = 1 identically.
11. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
Yes, this may be the case. Then the only time that you are likely to be asked to prove this trig identity is for a C1 paper. They don't currently teach complex numbers until you do further maths.
[I had to prove this trig identity using a right angled triangle in a C3 paper, and state which values of x the proof was valid for]

But we're not here to pass exams, we're here to enjoy maths! (at least I hope that's what the OP has been up to at Cambridge...)
12. Re: Prove cos2(x)+sin2(x) = 1
(Original post by Mark13)
I think this would be a reasonable question for a first-year undergraduate analysis course. In this case, attempting a proof by referring to right-angled triangles isn't going to be what's required.
Yes, you may be using different methods to prove this trig identity - especially if you are already at University.

But, there is nothing wrong with the proof that I suggested.

I have done a little bit of research. I also found a derivation for this in the following book.

A first course in Calculus 5/e. Serge Lang. Springer Verlag 1986.

This can be found on pages 136-137 of this edition of the book.

My argument would be, if its good enough for Serge Lang, it should be good enough for everyone else.
13. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
Yes, you may be using different methods to prove this trig identity - especially if you are already at University.

But, there is nothing wrong with the proof that I suggested.

I have done a little bit of research. I also found a derivation for this in the following book.

A first course in Calculus 5/e. Serge Lang. Springer Verlag 1986.

This can be found on pages 136-137 of this edition of the book.

My argument would be, if its good enough for Serge Lang, it should be good enough for everyone else.
You have COMPLETELY missed the point of this thread...

To quote the OP, "lets see how many different proofs we can get", so why on Earth are you on this particular thread ranting about one, short (and, as demostrated, less complete) proof of the identity as if it's all that this thread is for; and then going on to criticise one of the more proper versions?!

To quote the OP again, "well this is just for fun". The context is fun, and thus criticising the fact that multiple proofs have been posted is counter-intuitive, surely.
14. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
Yes, you may be using different methods to prove this trig identity - especially if you are already at University.

But, there is nothing wrong with the proof that I suggested.

I have done a little bit of research. I also found a derivation for this in the following book.

A first course in Calculus 5/e. Serge Lang. Springer Verlag 1986.

This can be found on pages 136-137 of this edition of the book.

My argument would be, if its good enough for Serge Lang, it should be good enough for everyone else.
There isn't anything wrong with the proof you suggested. It works. But it relies on a different definition of sin and cos to the one I used, and on pythagoras (I've actually seen pythagoras stated as s^2 + c^2 = 1 so those people would argue that your proof is circular).

I stated at the top of my proof "here's one from the power series definition". It's perfectly valid and hopefully welcome in a thread trying to collect as many proofs as possible. If you insist on making the proof as short as possible by freely assuming things that have already been proven, there's a pretty obvious one liner .

I think you'd have a fit if you did a real analysis course and found that you start out proving that (-1)*(-1) = 1 and other obvious things.
Last edited by SsEe; 29-08-2011 at 16:00.
15. Re: Prove cos2(x)+sin2(x) = 1
(Original post by SsEe)
There isn't anything wrong with the proof you suggested. It works. But it relies on a different definition of sin and cos to the one I used, and on pythagoras (I've actually seen pythagoras stated as s^2 + c^2 = 1 so those people would argue that your proof is circular).

I stated at the top of my proof "here's one from the power series definition". It's perfectly valid and hopefully welcome in a thread trying to collect as many proofs as possible. If you insist on making the proof as short as possible by freely assuming things that have already been proven, there's a pretty obvious one liner .

I think you'd have a fit if you did a real analysis course and found that you start out proving that (-1)*(-1) = 1 and other obvious things.
I've done a real analysis course, but that was very many years ago - possibly well before you were born.

Also note, that Serge Lang proves this very simply - that is (-1) * (-1) = 1.
Last edited by tonyhawken; 29-08-2011 at 16:04.
16. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
I've done a real analysis course, but that was very many years ago - possibly well before you were born.

Also note, that Serge Lang proves this very simply - that is (-1) * (-1) = 1.
Indeed it would have been. Well then you see that some "obvious" things actually require a bit of work if you boil your assumptions down to the basics.

Serge Lang clearly aren't working from the axioms for a complete, totally ordered field then.
Last edited by SsEe; 29-08-2011 at 16:06.
17. Re: Prove cos2(x)+sin2(x) = 1
(Original post by SsEe)
Indeed it would have been. Well then you see that some "obvious" things actually require a bit of work if you boil your assumptions down to the basics.

Serge Lang clearly aren't working from the axioms for a complete, totally ordered field then.
I am sure that a mathematician of Serge Lang's stature would use them if he felt they were necessary.
18. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
I am sure that a mathematician of Serge Lang's stature would use them if he felt they were necessary.
Well if his assumptions were nothing more than those axioms, it'd take him a few lines.

If his assumptions were nothing more than the ZF axioms of set theory.... it'd take him the best part of a lecture course to build up to it.

In that book you mentioned, he even talks about how 1>0 can be proven from the axioms. It's something we take for granted, but if you start out with a more basic set of assumptions, it's something that must be derived from them.

If the thread was "prove 1>0", you'd get some saying it was obvious, some proving it from the axioms of the real numbers, and one or two justifying it from the axioms of set theory. I guess you'd fall into the "it's obvious" camp. But worse, I think you'd still claim it was obvious if you were told that you couldn't assume anything beyond the axioms of the reals and that any claim had to be carefully proven. And that's where you're just wrong.
19. Re: Prove cos2(x)+sin2(x) = 1
(Original post by SsEe)
Well if his assumptions were nothing more than those axioms, it'd take him a few lines.

If his assumptions were nothing more than the ZF axioms of set theory.... it'd take him the best part of a lecture course to build up to it.

In that book you mentioned, he even talks about how 1>0 can be proven from the axioms. It's something we take for granted, but if you start out with a more basic set of assumptions, it's something that must be derived from them.

If the thread was "prove 1>0", you'd get some saying it was obvious, some proving it from the axioms of the real numbers, and one or two justifying it from the axioms of set theory. I guess you'd fall into the "it's obvious" camp. But worse, I think you'd still claim it was obvious if you were told that you couldn't assume anything beyond the axioms of the reals and that any claim had to be carefully proven. And that's where you're just wrong.
I suggest that you write to him. You may learn some mathematics. I do believe he use to be a contributor of Nicholas Bourbaki.
20. Re: Prove cos2(x)+sin2(x) = 1
(Original post by tonyhawken)
I suggest that you write to him.
How? He's been dead for many years!