I have a question concerning the following problem:
Let $B^{(1)}$ and $B^{(2)}$ denote two independent Brownian motions. Let $$W_t^{(1)} = \int_0^t \sin(s)dB^{(1)}_s + \int_0^t \cos(s)dB^{(2)}_s$$and let $$W_t^{(2)} = \int_0^t \cos(s)dB^{(1)}_s - \int_0^t \sin(s)dB^{(2)}_s.\\ $$ Show that $W_t^{(1)}$ and $W_t^{(2)}$ define two independent Brownian Motions.
I tried using Ito's formula for $f_1(x,t) = x\sin(t)$ and $f_2(x,t) = x\cos(t)$, but I don't really see where it is going. And for the independence I don't have an idea yet.
Thank you very much in advance!