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Calculus/PreCalc Proof

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Expert: - 11/14/2008


Question
Hi, I need a bit of help on starting this proof on my Honors Precalc HW. I have to prove that it is an identity and I don't know how to start, here is the identity:

I have to prove that
((1 + sinx + cosx)/(1 + sinx - cosx)) = ((1 + cos x)/(sinx))

If its okay could you help me start it off, not necessarily give me the anser all the way??  

Thanks,
Chels

Answer
Far side a, near side b, hypoteneuse c
(1 + a/c + b/c)/(1 + a/c - b/c) = (1 + b/c)/(a/c)

Put everything over c
(c/c + a/c + b/c)/(c/c + a/c - b/c)=(c/c + b/c)/(a/c)

Cancel the c's on top and bottom
(c+a+b)/(c+a-b)=(c+b)/a

Cross multiply
a(c+a+b) = (c+b)(c+a-b)
ac + a² + ab = c² + ca + cb + bc + ba - b²
ac + a² + ab = c² + ca + ba - b²

Rearrange the terms
a² + b² + ac + ab = c² + ac + ab

Note that both sides have an ac+ab, so subract that off
a² + b² = c²

I had to actually do it myself to see if it was really true, so now I'm done and it is indeed true (I've had a few questions that people asked me to prove that were not true).

If you want a little work to do, try to convert that back into trig functions in each step.  After that, you may have to move all of the conversions to the right side to make it a proper proof.

That may not be exactly what you asked for, but that's it for now.  Unless, of course, you ask me again.

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