You have to solve x = sin x
If you have a look at the graph of the sine function you see that it starts from the point (0,0). At this point the value of the sin function is zero and the value of x is also 0.
At no other point does x and sin x become equal.
Therefore the solution of the equation is x = 0.
There is nothing wrong with this equation.
In order to solve the equation, which is a transcedental one, you have to find the derivative of the function f.
The function f(x) = x - sin x (we've moved sinx to the left side)
We can note that is a monotone increasing function
( -1<cosx<1), so the difference 1 -cos x>0 =>f(x)>0, so f(x) is an injection.
We can also do a very simple calculus:
If f(x) is an injection, then x=0 is the only solution for x-sinx=0.