Bridging Trig Identities

Date: 06/07/2002 at 11:08:59
From: Susannah
Subject: trigonometric identities

Hi...please help. I have been staring at this question for ages and 
I'm getting nowhere:

  Prove
  tan 2A x sec A  = 2sin A  x sec 2A

My teacher told me that when proving these identities you have to
choose one side which is what I have been doing.  I started off by
opening up the identity 

  tan2A = 2tanA / 1 - tan*2 A 

and then that didn't get anywhere so I changed sides and opened up 

  sec2A

but that didn't help either. 

Thank you.

Date: 06/07/2002 at 17:05:15
From: Doctor Peterson
Subject: Re: trigonometric identities

Hi, Susannah.

Often when you are first working out a proof, it helps to work on 
both sides at once; then you can rewrite what you have done in a more 
proper way. This is sort of like constructing a bridge by building 
both ends toward the middle, but then driving across it in one 
direction.

In this case, as in many, it helps a lot to write everything in terms 
of sine and cosine:

    tan(2A) * sec(A) =? sec(A) * sec(2A)

    sin(2A)      1        1         1
    ------- * ------ =? ------ * -------
    cos(2A)   cos(A)    cos(A)   cos(2A)

(Notice how I indicated that the "=" is not yet known to be true; 
that's my way to keep myself honest when I work this way!)

Now you can apply a double-angle formula and do some canceling, and 
you will be able to make both sides look the same. Then you will have 
a real equation.

Now how do we walk across the bridge and make it a proper proof?

You will have written a sequence of equivalent equations of the form

    a =? b
    c =? d
    e =  e (this one doesn't need the "?"!)

As long as you have been working on each side separately, you can now 
say this:

    a   b
    =   =
    c   d
    =   =
    e = e

That is,

    a = c = e = d = b

and you have proved that a = b! Can you see now how we were working 
from both ends? We built the span on the left (a=c) and on the right 
(b=d) first, then the middle spans (c=e, d=e) and found we had 
connected the two shores. Then we just had to put it all in the right 
order and we were done.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/