Question

When x¡ú0 cosec(x) - cot (x)/ x =?

Answer 1

csc(x) = cot(x) /x
1/sinx = cosx/(xsinx)
1 = cosx /x , as long as sinx≠0
cosx = x

let's see what the graph looks like
if y = x and y = cosx

http://www.wolframalpha.com/input/?i=plot+y+%3D+x+%2C+y+%3D+cosx

looks like a solution around x = .65

these type of equations are hard to solve, since there is no "formula" way to do them.
A very good method is Newton's method, but I am not sure if you know Calculus

One simple way is to set your calculator to radians and start with some reasonable value for x

e.g.
x = .65, cos .65 = .796.. , ≠ x, so go higher
x = .7 , cos .7 = .76.48 , they are getting closer
keep nibbling away at the x
x = .75 , cos .75 = .7316.. , looks like I went too high
x = .74 , cos .74 = .738 , getting closer
x = .745 , cos .745 = .735
x = .747, cos .747 = .7337
let's go the other way:
x = .735, cos .735 = .7418..
mmmmhhh
x = .738, cos .738 = .7398..
x = .739, cos .739 = .739
not bad, correct to 3 decimals

did Newton's Method to get x = .739085..