# 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.