Proving Trigonometric Identities: 2cotx cscx - 1/1-cosx - 1/1+cosx

[thebeanstalk]

Hello,

I was stuck proving these given identities

*2cotx cscx - 1/1-cosx - 1/1+cosx "2cotx * cscx equals 1 over 1-cosx minus 1 over 1+cosx"

What I did here is:

I multiplty both sides by (1-cosx)(1+cosx) to get the denominator 1/1-cosx - 1/1+cosx so the answer is (1-cosx) + (1+cosx) all over (1-cosx)(1+cosx) and after that I was stuck

*1-cosx/1+cosx = (cscx - cotx)^2 "1-cosx over 1+cosx is equal to quantity cscx - cotx squared"

Hope this will get an answer. Thank you very much!

 

[Deleted member 4993]

Hello,

I was stuck proving these given identities

*2cotx cscx - 1/(1-cosx) - 1/(1+cosx) → What identity??

*(1-cosx)/(1+cosx) = (cscx - cotx)^2 → Where are you stuck? Show work!!

Hope this will get an answer. Thank you very much!

Please check your grouping symbols () and fix your original problem accordingly (otherwise, as posted, your problem does not make sense).

 

[thebeanstalk]

Sorry for the bad writings on the attached file.

 

[stapel]

I was stuck proving these given identities

*2cotx cscx - 1/1-cosx - 1/1+cosx "2cotx * cscx equals 1 over 1-cosx minus 1 over 1+cosx"

What I did here is:

I multiplty both sides by (1-cosx)(1+cosx) to get the denominator 1/1-cosx - 1/1+cosx so the answer is (1-cosx) + (1+cosx) all over (1-cosx)(1+cosx) and after that I was stuck

*1-cosx/1+cosx = (cscx - cotx)^2 "1-cosx over 1+cosx is equal to quantity cscx - cotx squared"

I don't understand the overlapping asterisks and quotemarks in the second line above...? I will guess that the first identity to prove is as follows:

. . . . .\(\displaystyle \mbox{1. Prove }\, 2\, \cot(x)\, \csc(x)\, =\, \dfrac{1}{1\, -\, \cos(x)}\, -\, \dfrac{1}{1\, +\, \cos(x)}\)

I don't understand what you mean by "multiplying both sides", since this is only a valid operation for equations (to be solved), not identities (to be proven). One must work on one side to prove it equal to the other. (here) A good first step in this case would be to convert the fractions on the right-hand side to the common denominator, combine, and see where this leads.

I will guess that the second identity to prove is as follows:

. . . . .\(\displaystyle \mbox{2. Prove }\, \dfrac{1\, -\, \cos(x)}{1\, +\, \cos(x)}\, =\, \left(\csc(x)\, -\, \cot(x)\right)^2\)

A good first step might be to expand the right-hand side and apply some identities.

If you get stuck, please reply showing all of your steps so far. Thank you!