Expert: Paul Klarreich - 2/9/2009

Question
So i'm having trouble solving this problem.

1. Prove the Identity
         Sin2x= 1/(tanx+cot2x)

Could you run me through the steps of solving. I keep getting stuck. It would really help to know where I went wrong. Thanks


Answer
Questioner:   brooke
Category:  Calculus
Private:  No
 
Subject:  proving an identity
Question:  So i'm having trouble solving this problem.

1. Prove the Identity
        Sin2x= 1/(tanx+cot2x)

Could you run me through the steps of solving. I keep getting stuck. It would really help to know where I went wrong. Thanks


================ NOTATION ==========================
I will write   s = sin x,  c = cos x, to save typing.
====================================================

Hi, Brooke,

For this, you will need to know:
1. The double angle identity for sin(2x),
1a. The double angle identity for cos(2x) (all of them)
2. The quotient identities.
3. How to simplify a complex fraction.

These were all part of your standard trig course, but you didn't really learn them until you got here.  [That's why your teacher has you doing this.]

Your example:
                 1
sin (2x) = ----------------
          tan x + cot 2x

                      1
2 sin x cos x = ----------------  << use 1.
               tan x + cot 2x


Now a standard practice, if you can't come up with something brilliant, is to change everything to sin x and cos x, then use a lot of algebra.


================ NOTATION ==========================
I will write   s = sin x,  c = cos x, to save typing. << I said that before,I know.
====================================================

              1
2 s c = ----------------     << use 2
       s/c + c(2x)/s(2x)


                  1
2 s c = --------------------
        s/c + (1 - 2s^2)/2sc  << use 1, 1a.

                 2 sc
2 s c = ------------------------------   << use 3: multiply by 2sc, the LCD
        2sc(s/c) + 2sc(1 - 2s^2)/2sc

              2 sc
2 s c = -------------------
         2s^2 + 1 - 2s^2

         2 sc
2 s c = --------  etc.
           1