Respiration in Organisms:
Respiration is the process by which organisms obtain energy from the breakdown of food molecules and release waste products like carbon dioxide. Different organisms have different respiratory systems to facilitate the exchange of gases.

Tracheal System:
The tracheal system is a specialized respiratory system found in insects, including cockroaches. It consists of a network of tiny tubes called tracheae that extend throughout the insect's body, delivering oxygen directly to the tissues. The tracheae open to the outside through small openings called spiracles, which are located on the sides of the insect's body segments.

Explanation:
Out of the given options, the organism that respires through a tracheal system is the cockroach. Let's take a closer look at each option to understand why the others do not respire through a tracheal system.

a) Earthworm:
Earthworms respire through their moist skin, which allows the exchange of gases with the environment. They do not possess specialized respiratory organs like tracheae or lungs.

c) Fish:
Fish respire through gills, which are specialized organs located on the sides of their body. Gills extract oxygen from water, allowing fish to respire.

d) Frog:
Frogs have lungs as their respiratory organs. They have a dual respiratory system, enabling them to respire both through their lungs and their moist skin. Lungs are used for respiration on land, while the skin plays a role in respiration in water.

Summary:
In summary, the organism that respires through a tracheal system is the cockroach. It uses a network of tracheae to deliver oxygen directly to its tissues. Other organisms, such as earthworms, fish, and frogs, have different respiratory systems adapted to their specific habitats and do not rely on a tracheal system for respiration.

Frogs have lungs, cockroaches have trachea, while fishes have gills for respiration. Earthworms don't have special organs for respiration, instead, they breathe through their skin. The tracheal system of respiration is present in insects.