proving trigonometric identities?

Question

i need help with my technique when i take on questions about this.

take this example:
prove that:

cos 2x / (1-sin 2x) = (1 + tan 2x) / (1-tan 2x)

all i ever do in these questions is start substituting things like
" sin^2 x + cos^2 x " in place of " 1 "

with no real ideas or aims with what i am doing, i just wade in blindly and hope it works.
so can anybody tell me any sort of techniques for dealing with questions like this?
like what's the first thing you always do with a question like this,
r there certain things you make sure to do that always help out?

i am sick of just charging blindly at these questions because i either get lucky and get the answer by what feels like a fluke, or more often i just make a more complicated mess than the original expression i started with.

can anybody help out? does not have to be with that example i gave up there, but whatever suits you.

cheers.

Answer 1

I don't have the answer but when I solve them, I generally begin by first substituting in that formula where I can, then I access the more difficult identities for things like sin(2x) and such.

A good list of all can be found at:
http://en.wikipedia.org/wiki/Trig_identities

you have to wade through some stuff but you'll get to them...
my first goal is to get everything in terms of just sin(x),cos(x), tan(x) and the other three. Then you manipulate them to simplify, i.e. factor out things, reduce (sinx/cosx=tanx) etc. then you should arrive at your answer clearly and brightly. While there is always wading through big mathematical messes in the middle, especially in this problem, eventually you will arrive at an answer you can be confident in.

Hope it helps...

Answer 2

(1 + tan 2x) / (1-tan 2x)
= (cos 2x + sin 2x)/(cos 2x - sin 2x), multiplied by cos 2x / cos 2x
= (cos 2x + sin 2x)(cos 2x - sin 2x)/(cos 2x - sin 2x)^2
= cos 4x / (1 - sin 4x)

You must have had a typo.
cos 2x / (1-sin 2x) = (1 + tan x) / (1-tan x) Is this what you want?

Ideas:
1) Read examples carefully.
2) Learn the most important tricks, copy them and review later so that they become yours.
3) Do more practice.
4) When you try to prove a trig. identity, go from the side which you are less familiar with to the side which you feel more comfortable.

Answer 3

I always tackle the most complicated side first.
I try to convert everything to sines and cosines.

Answer 4

You are right that it is equal to tan(2x) . now tan(2x) = tan(x+x) =(tanx + tanx) /(1 - tanx tanx) =2tanx /(1 - tan(x) tanx or 2tanx /1 - tan^2(x)