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April 21st, 2018, 06:39 AM  #1 
Newbie Joined: Apr 2018 From: East London Posts: 12 Thanks: 0  Determine values of variables for each of the following identities
1. 2x / Cosx Sinx = Cosx + Sinx 2.1  Sin 2x / Sinx Cosx = Sinx  Cosx 3.Sinx + Sinx /1 + Cosx + Cos2x = Tanx Last edited by greg1313; April 21st, 2018 at 01:36 PM. 
April 21st, 2018, 07:10 AM  #2 
Senior Member Joined: Oct 2009 Posts: 770 Thanks: 276 
2. $$1\frac{\sin(2x)}{\sin(x)}  \cos(x)=\sin(x)\cos(x)$$ if and only if $$1\frac{2\sin(x)\cos(x)}{\sin(x)} = \sin(x)$$ if and only if $$12\cos(x) = \sin(x)$$ Letting $$t=\tan(x/2)$$, we get $$12\frac{1t^2}{1+t^2} = \frac{2t}{1+t^2}$$ if and only if $$1+t^2  2+2t^2 = 2t$$ if and only if $$3t^2  2t1=0$$ Discriminant is $$D=7$$ So $$t = \frac{2\pm \sqrt{7}}{6}$$ So $$x = 2\text{arctan}\left(\frac{2\pm \sqrt{7}}{6}\right)$$ Now I'm sure you can do the rest in the same way! Last edited by skipjack; April 26th, 2018 at 08:28 AM. 
April 21st, 2018, 10:00 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,464 Thanks: 2038 
The second equation lacked parentheses; it should have been $\frac{1  \sin 2x}{\sin x  \cos x} = \sin x  \cos x$.

April 21st, 2018, 01:38 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond  

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determine, identities, values, variables 
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