Expert: Paul Klarreich - 3/1/2007

Question
Prove the identity:  sin³x-cos³x/sinx+cosx=csc²x-cotx-2cos²x/1-cot²x  

Answer
Questioner:   Justin
Category:  Advanced Math
 
Subject:  Trig
Question:  Prove the identity:  sin³x-cos³x/sinx+cosx=csc²x-cotx-2cos²x/1-cot²x
.........................................................
Hi, Justin,

I assume your identity is:

WARNING: THIS DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL DIFFICULT TO VIEW ON CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

sin^3(x) - cos^3(x)   csc^2(x) - cot x - 2 cos^2(x)
------------------- = -----------------------------
  sin x + cos x          1 - cot^2(x)

In the following, I will change all the stuff on the right side to sines and cosines,

and write  S, C for those, to simplify the algebra.

 1       C    2C^2
------ - --- - ----
S^2      S      1
----------------------
      C^2    
  1 - ---
      S^2

Clear fractions, using an LCD of S^2:
                      
1  -  CS - 2C^2S^2
----------------------
    S^2 - C^2    
OK, that's the right side.  Now work on the left side:

S^3 - C^3
---------
S + C

Factor the top:

(S - C)(S^2 + SC + C^2)
-----------------------
  (S + C)

(S - C)(1 + SC)
----------------
  (S + C)

Multiply out:

S - C + S^2C - SC^2
-------------------
  (S + C)

S - C + SC(S - C)
------------------
  (S + C)

Multiply top and bottom by (S - C), to make the bottom match the right.

[S - C + SC(S - C)](S - C)
--------------------------
(S + C)(S - C)

Multiply out on top.
S^2 - 2SC + C^2 + SC(S^2 - 2SC + C^2)
-------------------------------------
S^2 - C^2

Locate occurrences of  s^2 + c^2

1 - 2SC + SC(1 - 2SC)
----------------------
S^2 - C^2

1 - 2SC + SC - 2S^2C^2
----------------------
S^2 - C^2

1 - SC - 2S^2C^2
----------------------
S^2 - C^2

That's it.  Identical to the right side.

WHEN I COME WITH SOMETHING BRILLIANT, I'll let you know.