Oh geez. This is going to be a long one. Here I go:
There's 2 ways to do by limits. As x -> c and as delta x (for lack of sign I'll use #) -> x. So with signs as #x -> x. I'll do the #x method because I find it easier to simplify. It can be even easier once you start using Power, Product, Quotient, and Chain rules.
So:
lim ........................... 1 + csc(#X + X) ................... 1 + cscX
#X->0 ...................... ---------------- ..... - ........ -----------
................................1 - csc(#X + X) ...................... 1 - cscX
...............................-------------------------------------------
...........................................................#X
lim.................................... (1 - cscX)(1 + csc(#X+X)) - (1 - csc(#X+X))(1 + cscX)....................... All that I've done here is common denominators
#X->0.............................. -------------------------------------------------------................................... followed by simplification.
.........................................................#X(1 - csc(#X+X))(1 - cscX)
lim....................................1 + csc(#X+X) - cscX - (cscX)(csc(#X+X)) - 1 - cscX + csc(#X+X) + (cscX)(csc(#X+X))
#X->0..............................------------------------------------------------------------------------------------------................Distribute.
.........................................................................#X(1 - csc(#X+X))(1 - cscX)......................................................I've already taken care of distributing the -1.
lim..........................................................2csc(#X+X) - 2cscX
#X->0................................................----------------------------.............Simplify
.........................................................#X(1 - csc(#X+X))(1 - cscX)
lim....................................................2(csc(#X+X) - cscX)
#X->0............................................-----------------------.............Simplify more (it's always good in a hard problem like this; I just use rules)
.......................................................#X(1 - csc(#X+X))(1-cscX)
Taken out of context to simplify:
...................................1...............................................1...........................
...................------------------------......-........--------------------------........ Rule for simplifying sin(A+B)
...................sin#X cosX + cos#X sinX..............................sinX........................
............................sinx - sin#XcosX - cos#XsinX
...................--------------------------------------
......................sinXsin#XcosX + cos#X(sinX)^2
Alright dude, screw this we're doing rules, as proving trig stuff is really nasty. Even my calc teacher said so. After tons of simplification you should get:
lim....................................................................2cscX
#X->.............................................................-----------.........................The #X on the side needs to be cancelled off so you get:
.........................................................#X(1 - csc(#X+X))(1 - cscX)
lim....................................................................2cscX
#X->.............................................................-----------.........................Plug 0 for #X
........................................................ (1 - csc(#X+X))(1 - cscX)
lim....................................................................2cscX
#X->.............................................................-----------.........................And there's your answer.
..............................................................(1 - cscX)(1 - cscX)