Alex,
The problem definition says that t_1 is sufficient for \theta_1 when \theta_2 is known. Once theta_2 is known, it is a constant like any other and can be used as part of a statistic. Once theta_2 is known the distribution reduces to a single parameter distribution depending only on theta_1. As long as I define my statistic after the distribution is given to me, all is well (or so I think). If I were to define the statistic without the knowledge of \theta2, that is when the problem starts.