Solve the equation x - sinx = 0.

This  is a transcedental equation and for solving it we have to differentiate the function.

f'(x)=1-cosx

The derivative 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'll calculate f(0):

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

Because f(x) is an injection, that means that f(x) is increasing, so, x=0 is the only solution for  x-sinx=0.