The single machine coupled task scheduling problem includes a set of jobs, each with two separated tasks and there is an exact delay between the tasks. We investigate the single machine coupled task scheduling problem with the objective of minimizing the makespan under identical processing time for the first task and identical delay period for all jobs, and the time-dependent processing time setting for the second task. We provide optimal solution under certain conditions, and propose a heuristic for the general case. The numerical study shows that the heuristic performs very well.