It is possible to solve the equation x = sin x?

Yes, it is possible to solve the equation, using
derivative of the function associated to the
expression:

f(x) = x -
sinx

The function is a continuous function, formed by
elementary functions as the linear one,x , and the trigonometrical one, sin x, so we can
calculate it's derivative.

f'(x) = (x -
sinx)'

f'(x)=1-cosx

We notice
that the first derivative is an increasing function.

(
-1<cosx<1), so the difference 1 -cos x>0 =>f(x)>0, so
f(x) is one-one function.

We can also do a very simple
calculus:

f(0)=0-sin0=0-0=0.

Because
f(x) is an one-one function, x=0 is the only solution for the
equation x-sinx=0.