#Trigonometry proof question

1 messages · Page 1 of 1 (latest)

fresh whale
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Tried with a few different methods, even tried to prove by induction (can't find a base case), but I can't figure out a method for this. If anyone can lend a hand I'd appreciate it

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long pierBOT
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dusky osprey
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And this identity is meant to hold for all real (or possibly even complex) values of theta.

fresh whale
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I learnt induction two days ago I was just getting a bit desperate for a solution to be honest

dusky osprey
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Wait, why are they teaching you induction two days before trigonometry?

fresh whale
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I do further maths and maths (different subjects in the uk), so I learn different things at the same time

dusky osprey
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That kind of feels like something they shouldn't let you do because it'd be confusing? Like, it feels like maths should be a prereq of further maths.

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Either way.

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The first step in proving any trig identity, I find, is generally converting it into purely sine and cosine.

fresh whale
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I did convert the tanx into sinx/cosx, but I couldn't find any progression further than that

dusky osprey
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Well, I feel like there's an obvious next step after that, right?

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Like, show the work you've done so far.

timber jay
dusky osprey
fresh whale
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I'll write the working I've done so far

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so I've taken sin^2(x)/(cosx+cos^2(x)) from the sinx/cosx identity

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I'm just uncertain where to go from here

dusky osprey
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Do you recall the Pythagorean identity?

fresh whale
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Yes

dusky osprey
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You should always be thinking about that whenever you see sine or cosine is squared.

fresh whale
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so take the numerator as 1-cos^2(x)?

dusky osprey
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That's certainly an idea.

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And what can we tell about 1 - cos^2(x)?

fresh whale
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Could we take it as difference of squares

dusky osprey
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Yes.

fresh whale
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Thank you I've got it

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I really appreciate your help

dusky osprey
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No problem.

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In general, when you're proving an identity.

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The goal is to transform one side into the other side.

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But you can work from both ends.

fresh whale
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Oh right I see

dusky osprey
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Like, you can begin by assuming that the equation is true, not to prove it circularly, but to explore its implications to find a path to proving it.

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I noticed 1/cos(x) - 1 = (1 - cos(x))/cos(x), and that if we cross-multiply with tan(x)sin(x)/(1 + cos(x)), that we'd get the standard Pythagorean identity sin^2(x) = 1 - cos^2(x).

fresh whale
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I see

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I wouldn't have thought of that approach I appreciate seeing how you think of it

dusky osprey
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And that of course doesn't prove the identity, but it shows that the way to prove it might involve the Pythagorean identity and the difference of squares.

fresh whale
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Oh right

dusky osprey
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Though I guess technically all the algebra we did from tan(x)sin(x)/(1 + cos(x)) = 1/cos(x) - 1 to sin^2(x) = 1 - cos^2(x) is reversible.

fresh whale
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I can see that yeah

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Is there any way I can give you any praise or do I just need to close this?

dusky osprey
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When you close it you'll get the option to thank me.

fresh whale
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Thank you once again

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+close

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# fresh whale +close
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# long pier

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