Math help - Verifying Trigonometric Identities

Filbert

Filbert

CAGiversary!
*sigh* It had been years since I had a math course, but I'm currently taking a Trigonometry course so I can get into some Physics classes. Well the homework assignments usually only take me an hour or two. Then we got to Verifying Trigonometric Identities.

I always had trouble with factoring and proofs. I've now spent about eight hours on this handful of questoins. Four eventually fell, but one is being particularly stubborn.

[ ( secX - tanX )^2 + 1 ] / [ secX ( cscX ) - tanX ( cscX ) ] = 2 ( tanX )

I'm not looking for someone to do my work for me (I like to uderstand why something works before I put it to use), but I just don't know what to do about this one. I'd post the stuff I've done so far but it's so ridiculously long compared to the other problems in the homework that I'm sure something could be wrong at any point.

Any help is greatly appreciated.
 
If anybody cares I met with my professor twice while he tried to clue me into what I was doing wrong. He first told me that the identity could be verified without changing everything to sin and cos. That apparently was my first problem since I was changing everything to sin and cos immediately. Doing so turned the left side into:

( cos^2X + 1 ) / [ secX ( cscX ) - tanX ( cscX ) ]

then eventually to:

cos^3 ( sinX ) - cos^2X ( sinX / cosX ) sinX + cosX ( sinX ) - ( cosX / sinX ) sinX

And I went downhill from there. After trying again without changing everything to sin and cos I got much closer. I ended up with something like:

2 [ ( -tanX sinX - tanX sin^2X ) / cosX ] = 2 ( sinX / cosX )

Which seemed to be along the right path, but I was sure I had made a mistake somewhere. I went back to see him again right before class and he quickly pointed out that I had an incorrect sign in my first step. I went right out in the hall then and did the problem as follows:

[ ( secX - tanX )^2 + 1 ] / [ secX ( cscX ) - tanX ( cscX ) ] = 2 ( tanX )
[ sec^2X - 2 secX tanX + tan^2X + 1 ] / [ secX ( cscX ) - tanX ( cscX ) ] = 2 ( tanX )
[ 2 sec^2X - 2 secX tanX ] / [ cscX ( secX - tanX ) ] = 2 ( sinX / cosX )
[ 2 secX ( secX - tanX ) ] / [ cscX ( secX - tanX ) ] = 2 ( secX / cscX )
2 ( secX / cscX ) = 2 ( secX / cscX )

It's strange how simple the problem seems now. I really should have stopped myself a lot earlier and made sure I was doing my first two steps correctly before I let the problem grow to abnormal length.
 
I tried to tackle this, but I didn't know (secx)^2 + (tanx)^2 + 1 = 2(secx)^2, so I stalled midway through. I also initially tried the sine/cosine thing you did.

But yeah, that's pretty straightforward in hindsight,
 
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