Hi:
I am asked to prove the following identities:
a) 2cot2θ≡cotθ-tanθ
b) tan3θ≡3tanθ-tan³θ/1-3tan²θ
I think I should tackle the most complicated side as it is easier to aim for the easier looking side. (hope that makes sense )
I should warn you that I'm not good at this so please excuse my fumbling
a) 2cot2θ≡cotθ-tanθ
I will attempt the RHS, cotθ-tanθ
cotθ-sinθ/cosθ
so I'm stuck! Told you
Hi:
thank you:
I was aware of cotθ=1/tanθ=cosθ/sinθ
But I find this a mine field. And never know where to start nor which replacements to use. It's really confusing to me.
I would like to return to this a little later and ask you some questions about your thinking, if I may?
But first:
b) tan3θ≡(3tanθ-tan³θ)/1-3tan²θ
so the LHS has only one thing to play with. So I should tackle this side?
tan3θ=sin3θ/cos3θ
On the right track?
I agree.
Also thank goodness for LaTeX. It's not that hard to write on paper though
Write
Note that and
Hence
Subbing in our double angle identity for \tan wherever occurs:
Looking at the numerator:
Get the same denominator by multipling by
Distribute the and simplify:
Note the similarity of this to the right hand side's numerator. Therefore, leave it like this and work on the denominator
Looking at the denominator (of the overall fraction)
Rewrite 1 as
The second term can be simplified to
Since the two terms now have the same denominator we can combine them:
The numerator of this fraction can be simplified to
As this is also similar to the RHS leave it in this form.
Bringing both parts together
Numerator:
Denominator:
Which makes our fraction equal to
Dividing by a fraction is the same as 'flipping' it (taking the reciprocal) and multiplying.
cancels to leave us with:
Well. That was a rather bigger than expected process.
Thanks for that
Your right - I should learn Latex, and will make an effort to do so after this weekend, which is this coming Monday. Promise Promise.
For anybody reading this who is interested, I just did a search on Latex video tutorials and came up with a couple of sites instantly. It seems there is a lot of help available online, as always.
TechScreencast - LaTeX Intro :: tutorial, videos, articles, screencast to learn technology"
latex video tutorial rapidshare, megaupload, torrent download
Though I haven't looked closely at them, yet, I'm pretty certain they will be helpful and are reputable sites.
Is the last link ok?