June 27th, 2019, 09:43 AM  #1 
Newbie Joined: Jan 2017 From: London Posts: 10 Thanks: 0  Reconcile 2 Trigonometric Indentities
Hi All, I'm stuck on a particular problem set: Establish the validity (reconcile) of the following  (1+sin x)/cos x = cos x/(1sin x) I'm finding it difficult to walk from either left to right or right to left i.e. I cannot decompose into other identities that will lead me to the other side. I thought about expanding the left side to sec x + tan x but that leads me nowhere. I'm stuck!! Help!! Thanks Last edited by skipjack; June 28th, 2019 at 11:05 PM. 
June 27th, 2019, 09:53 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390 
just get both sides over a common denominator and it should be obvious.

June 27th, 2019, 10:12 AM  #3 
Senior Member Joined: Jun 2019 From: USA Posts: 125 Thanks: 41  
June 27th, 2019, 10:23 AM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390  
June 28th, 2019, 08:37 AM  #5 
Senior Member Joined: Jun 2019 From: USA Posts: 125 Thanks: 41  
June 28th, 2019, 08:43 AM  #6 
Newbie Joined: Jan 2017 From: London Posts: 10 Thanks: 0 
Thanks @romsek @DarnItJimImAnEngineer. Making the denominator common amongst both and crossing out common terms does make it obvious. I also took another path; I split (1sin x) into (1sin^2x)/(1+sin x) on the right hand side. Then cos x/[(1sin^2x)/(1+sin x)] (using the identity cos^2x+ sin^2x =1) and one can simplify this to what's on the left = (1+sin x)/cos x Last edited by skipjack; June 28th, 2019 at 10:33 AM. 
June 28th, 2019, 11:09 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2205 
Does that always work?

June 28th, 2019, 06:51 PM  #8 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
$\displaystyle \frac{1+\sin x}{\cos x}\cdot\frac{1\sin x}{1\sin x}=\frac{1\sin^2x}{\cos x(1\sin x)}=\frac{\cos^2x}{\cos x(1\sin x)}=\frac{\cos x}{1\sin x}$

June 28th, 2019, 11:06 PM  #9 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2205 
What exactly does that prove?
Last edited by skipjack; June 28th, 2019 at 11:30 PM. 
June 30th, 2019, 02:31 PM  #10 
Newbie Joined: Jan 2017 From: London Posts: 10 Thanks: 0  

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indentities, reconcile, trigonometric 
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