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### Bridging Trig Identities

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

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/
```
Associated Topics:
High School Trigonometry

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