Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion, i.e. \(\left(b_{t}ight)_{t \geqslant 0},\left(\beta_{t}ight)_{t \geqslant 0}\) are independent one-dimensional Brownian motions. Show that for all \(\sigma_{1}, \sigma_{2}>0\)\[W_{t}:=\frac{\sigma_{1}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}} b_{t}+\frac{\sigma_{2}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}} \beta_{t}, \quad t \geqslant 0\]

is a one-dimensional Brownian motion.