Tinea Versicolor: This is a common fungal infection of the skin caused by Malassezia yeast, which is naturally found on the skin. It often presents as a rash on the trunk and can be resistant to treatments like ketoconazole shampoo if not used correctly or in sufficient duration. The lack of improvement with prednisone (which is more effective against inflammatory conditions) and ketoconazole shampoo suggests a possible misdiagnosis or insufficient treatment of tinea versicolor.

Psoriasis: Although prednisone is typically effective for psoriasis, the dose or duration might have been insufficient. Psoriasis can present with various types of rashes, including on the trunk.
Seborrheic Dermatitis: This condition can cause a rash on the trunk and might not fully respond to ketoconazole shampoo, especially if the shampoo is not used as directed or if the condition is severe. Prednisone might help, but again, the dose or duration could be factors.
Pityriasis Rosea: This is a skin condition that often starts with a single patch (herald patch) followed by an eruption of smaller patches, typically on the trunk. It can last for several weeks to a few months and does not respond to prednisone or ketoconazole.

Cutaneous T-Cell Lymphoma (Mycosis Fungoides): This is a rare type of skin cancer that can mimic various dermatoses, including eczema or psoriasis. It's crucial to consider this diagnosis, especially if the rash is persistent and not responding to typical treatments.
Cutaneous Lupus: Systemic lupus erythematosus (SLE) can cause skin rashes, and cutaneous lupus can present with lesions on the trunk. The lack of response to prednisone might indicate a need for a different or more aggressive treatment approach.
Sarcoidosis: This condition can cause skin lesions, among other symptoms, and might not respond to standard treatments for dermatological conditions.

Lichen Planus: While more commonly affecting the mucous membranes, skin, nails, and hair, lichen planus can present with a rash on the trunk. It's less likely but should be considered in the differential diagnosis due to its potential for significant morbidity if not treated.
Dermatomyositis: An inflammatory disease characterized by skin rash and muscle weakness, dermatomyositis can present with a rash on the trunk. It's a rare condition but important to diagnose due to its association with underlying malignancy in some cases.