Expert: Paul Klarreich - 12/11/2011

Question
prove that (cscA+cotA)/(cscA-cotA) = (1+2cosA+cos^2A)/(sin^2A)

Answer
Questioner: Ashley
Country: Ontario, Canada
Category: Advanced Math
Private: No
Subject: Trig Identity
Question: prove that (cscA+cotA)/(cscA-cotA) = (1+2cosA+cos^2A)/(sin^2A)
..........................................................................
LHS:

csc A + cot A
--------------
csc A - cot A

Apply reciprocal and quotient identities:

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

1/s + c/s
----------
1/s - c/s

Clear fractions; multiply all by s:

1 + c
------
1 - c

Rationalize the denominator:

1 + c  1 + c
-----  -----
1 - c  1 + c

(1 + c)^2
----------
1 - c^2

I think you can finish up from here.