how do you prove the identity: csc²(X/2) = (2secX)/ (secX-1)?

Question

thanks micheal for answering...but when proving identities...you are only allowed to work on one side. if you work on both sides...you are automatically asumming that the equation is true

Answer 1

You can work with both sides if you dont change any part of that,

Let's work with the first side...
csc^2 (x/2) = 1 / sen^2(x/2)

sen(x/2) = square root (( 1 - cos x)/2)
sen ^2 (x/2) = (1-cosx) /_2

csc^2 (x/2) = 2 / (1-cosx)

and the other side (this proves that 2 / 1-cosx is the same as the right side of your identity)

2(1/cosx)
------------
(1/cos x) -1

2/cosx
---------
(1-cosx)/cosx

= 2 / (1-cos x)

Join both sides and you have proved the identity

Answer 2

csc²(X / 2) = (2secX) / (secX-1)
use identities to convert to just sin and cos
1 / sin^2(x / 2) = (2 / cos(x)) / (1 / cos(x) - 1)
simplify fraction on right.
1 / sin^2(x / 2) = 2 / (1 - cos(x))
remove all fractions by cross multiplication
1 - cos(x) = (2)(sin^2(x / 2))
use sin half angle identiy
1 - cos(x) = (2)(1/2)(1 - cos(x))
1 - cos(x) = 1 - cos(x)
1 = 1

Answer 3

WTF? What is that??