 Solution

The process $Y_t=B_{T-t}-B_T$ is centered Gaussian and continuous. Its covariance isas follows by expanding Brownian covariances, or by reading its increments backwards. The Gaussian-process characterization of Brownian motion therefore shows that $(Y_t{)}_{0\le t\le T}$ has the same law as $(B_t{)}_{0\le t\le T}$.