Proving Reciprocal Identities

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1. Feb 15, 2017

Schaus

1. The problem statement, all variables and given/known data
(secx+1)/(sin2x) = (tanx)/2cosx-2cos2x)

2. Relevant equations

3. The attempt at a solution
Left Side
((1+cosx)/cosx)/2sinxcosx

((1+cosx)/cosx) x (1/2sinxcosx)
cancel the a cosx from both to get
(1/2sinxcosx)
This is all I could manage with left side so I tried right side
Right Side
(sinx/cosx)/2cosx-2cos2x)
I'm stuck here. I've been trying to find something to change the denominator of the Right Side but I can't think of anything that will work. If someone could let me know where I am going wrong it would be greatly appreciated!

2. Feb 15, 2017

haruspex

What happened to the 1+cos?

Rather than working each side separately, multiply out to get rid of all the denominators.

3. Feb 15, 2017

Schaus

I thought I could cancel a cosx, maybe I cannot. I tried to eliminate the denominator like you said. Here's what I got.
(secx+1)(2cosx-2cos2x) = (Sin2x)(tanx)

((1+cosx)/cosx)(2cosx-2cos2x)=(2sinxcosx)(sinx/cosx)
Expanding
((2cosx-2cos2x+2cos2x-2cos3x)/cosx) = 2
Moving the cosx under left side to the right side and simplifying
2cosx-2cos3x = 2cosx
Does this look right? And if so, where do I go from here?

4. Feb 15, 2017

haruspex

Check the right hand side.

5. Feb 15, 2017

Schaus

Woops. I think I should have gotten 2sin2xcosx
2cosx-2cos3x = 2sin2xcosx
Does this look right?

6. Feb 15, 2017

haruspex

Yes. Keep simplifying.

7. Feb 15, 2017

Schaus

2cosx-2cos3x = 2sin2xcosx
2cosx(1-cos2x) = 2sin2xcosx
2cosx(sin2x) = 2sin2xcosx
2cosxsin2x = 2sin2xcosx
Does this work?

8. Feb 15, 2017

haruspex

Yes.
Of course, it is not strictly kosher to start with the thing to be proved and deduce a tautology. You need all the steps to be reversible. They are in this case, but it is cleaner to rewrite it in the more persuasive sequence: start with the tautology and deduce the thing to be proved.

9. Feb 15, 2017

Schaus

So I should start with 2sin2xcosx and work backwards? Thank you for all the help by the way.

10. Feb 15, 2017

haruspex

Ideally, yes.

11. Feb 15, 2017

Schaus

Ok, I'll try it. I'm going to have to practice these quite a bit more I think.