Consider the following process:
$$X_t=\sin^2(W_t) + t,$$ where $W$ is Brownian Motion.
First define $f(t, x):= \sin^2(x) +t$.
This yields $$ dX_t=df(t, W_t)= f_t(t, W_t) dt + f_x(t, W_t)dW_t + \frac{1}{2}f_{xx} (t, W_t) dW_td W_t $$
$$= 1 dt + 2 \sin W_t \cos W_t dW_t + (\cos^2 W_t -\sin^2 W_t) dt$$ $$ = (1+ \cos^2 W_t -\sin^2 W_t ) dt + 2 \sin W_t \cos W_t d W_t$$
Is it formally ok? The characteristic pair would be $(1+ \cos^2 W_t -\sin^2 W_t, 2 \sin W_t \cos W_t )$
Is this the right interpretation?