The Shepherd took the tired traveler inside the cave.

Explanation:
The tired traveler was taken inside the cave by the Shepherd. Let's explore the reasons behind this choice and understand why the other options are not suitable.

a) Inside a small cottage:
A small cottage might provide shelter and comfort, but it may not be readily available in the vicinity. Moreover, the Shepherd might not have access to a cottage nearby, especially if they are traveling in a remote area. Therefore, this option is less likely.

b) Inside a palace:
A palace is a grand residence of a monarch or a noble. It is highly unlikely that a Shepherd would have access to a palace or that there would be a palace nearby. Palaces are usually associated with luxury and not commonly found in rural or remote areas. Hence, this option is unlikely as well.

d) Inside his hut:
A hut is a small, simple dwelling typically made of natural materials like wood, mud, or thatch. While it is more plausible for a Shepherd to have a hut compared to a palace or a cottage, it may not be suitable for accommodating a tired traveler. Huts are usually small and may not provide enough space or comfort for a weary traveler. Therefore, this option is also less likely.

c) Inside the cave:
A cave, on the other hand, can serve as a natural shelter and protection from the elements. Caves are often found in hilly or mountainous regions and can provide a safe haven for travelers, especially when there is no immediate access to buildings or shelters. The Shepherd might have chosen to take the tired traveler inside the cave to offer them a place to rest and recuperate.

In conclusion, the Shepherd took the tired traveler inside the cave because it was a feasible and practical option for providing temporary shelter and respite from the journey.