Let $W_t$ denote a Brownian Motion, use ito's formula to compute the differential of the following stochastic process: $$X_t=e^{t/2}\cos(W_t)$$
Ito's formula is supposed to be used when we know a diffusion differential equation like $dx=\mu dt+\sigma dW_t$, then we can apply it to solve a $dG(x,t)$ by: $$dG=\left(\frac{\partial G}{\partial t}+\frac{\partial G}{\partial x}\mu+\frac{1}{2}\frac{\partial^2G}{\partial x^2}\sigma^2\right)dt+\frac{1}{2}\frac{\partial G}{\partial x}\sigma dW_t.$$ But for this question, why we can solve the $dX_t$ directly?
Also, the answer to this question suggests that we should use the following Ito's formula to solve $dX_t$: $$dX(t,W_t)=\frac{\partial X(t,W_t)}{\partial t}dt+\frac{\partial X(t,W_t)}{\partial W}dW_t+\frac{1}{2}\frac{\partial ^2 X(t,W_t)}{\partial W^2}dt$$
What's the logic behind this Ito formula?