**Core Concept:** The question is testing the understanding of a rare disease manifestation and their association with certain disorders. It is crucial for students to recognize the characteristic features of these diseases, as they may often appear similar to common diseases but have unique implications.

**Why the Correct Answer is Right:**
The correct answer, **C. Leprosy**, is a chronic bacterial infection caused by Mycobacterium leprae or Mycobacterium lepromatosis. The disease primarily affects the skin, peripheral nerves, and mucous membranes. In this case, the patient presents with arthritis, hyperpigmentation of skin, and hypogonadism.

Arthritis is a common symptom in leprosy due to nerve damage and inflammation. Hyperpigmentation is caused by increased melanin production in response to the chronic inflammation. Hypogonadism results from the destruction of the hypothalamic-pituitary-gonadal axis, leading to reduced sex hormone production.

**Why Each Wrong Option is Incorrect:**
A. **Lepromatosis** (Option B) is a misconception as the term "lepromatosis" was coined to describe a severe form of leprosy. However, the correct answer is Leprosy (Option C).

B. **HIV infection** (Option D) is not the correct answer as HIV primarily affects CD4+ T cells, leading to opportunistic infections and neurological complications. Leprosy affects the peripheral nerves and skin, but not the hypogonadism seen in HIV.

**Clinical Pearl:** This question emphasizes the importance of considering rare diseases like leprosy when evaluating patients with unusual symptoms. It is essential for students to have a broad differential diagnosis in mind and consider the context of the patient's history, clinical presentation, and diagnostic tests to confirm the diagnosis.