Prove that sinx=x for x is a real number has only one solution at x=0

Answer 1

on the same grid, graph both
y = x, which is a straight line, with slope 1
y = sinx

http://www.wolframalpha.com/input/?i=plot+y+%3D+x,+y+%3D+sinx

What do you notice?
Where do they intersect ?

slope of y = sinx
= dy/dx = cosx

when x = 0, cos0 = 1
for any value of -π/2 < x < p/2 , cosx < 1
so the graphs rise at different rates and will never meet again.