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Okay, here it is, that really annoying trig identity question...

sinC + cosC - 1
--------------- ............................................... =
sinC - cosC + 1
cosC
-------
sinC + 1
[sinC + (cosC - 1)] [sinC + (cosC - 1)]
---------------------------------------
[sinC - (cosC - 1)] [sinC + (cosC - 1)]

sin^2C + sinCcosC - sinC + sinCcosC + cos^2C - cosC - sinC - cosC + 1
---------------------------------------------------------------------
sin^2C + sinCcosC - sinC - sinCcosC - cos^2C + cosC + sinC + cosC - 1

which simplifies to...
2 + 2sinCcosC - 2sinC - 2cosC
-----------------------------
sin^2C - cos^2C + 2cosC - 1

2(cosC - 1)(sinC - 1)
---------------------
2(cosC - cos^2C)

which then simplifies to...
1 - sinC
---------
cosC

multiply both sides by cosC to get...
cosC(1 - sinC)
--------------
cos^2C

and since cos^2C = (1 - sin^2C), we stick this into the denominator...
cosC(1 - sinC)
--------------
(1 - sin^2C)

and since (1 - sin^2C) = (1 + sinC)(1 - sinC), we can elinate (1 - sinC) from both sides and obtain:
cosC
------
1 + sinC


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