Show that there are at least two values of x

Question

in the interval (−π,π) such that x=1+sin(x)

Answer 1

A rough sketch of the two functions
y = x , and y = 1+sinx will show that y = x clearly crosses y = 1 + sinx only once. So your statement is not correct.

evidence:
http://www.wolframalpha.com/input/?i=solve+x%3D1%2Bsin(x)