Expert: Robi Bhattacharjee - 8/19/2008

Question
Dear Richard Reisner
I have a problem with a trignometric proof. I tried my best but can not be able to prove. Can you help me? Proof is:
1/(Csc a -Cot a)-1/Sin a =1/Sin a -1/(CSc a+Cot a)

Answer
Hello Hafiz,

You could start at the left side of this equation and start simplifying, but that would be hard and complicating. So instead, let us notice that if we prove

2/sina = 1/(Csc a - cota) + 1/(csc a + cota)


then we can easily show that your problem is true by subtracting 1/sina and 1/(Csca + Cota) from both sides.


So now, to prove the equation i came up with, let us start with the RIGHT side and start simplifying,

1/(csc a - cota) + 1/(csca + cot a)

=        csca + cota + csca - cota
        -------------------------
       (csca - cota)(csca + cota)

=           2 csca
       ---------------                 using difference of squares
        (csca^2 - cota^2)

=   2csca
  -------                              since cota^2 + 1 = csca^2
     1

= 2csca

=  2/sina         since csca = 1/sina

this means your original statement is true.

Sorry about the earlier answer, something got screwed up in the formatting.

Sorry that no one bothered to answer this earlier.

I hope this helped you out,

Robi

Feel free to ask