Final answer:
Due to unclear elements in the original question, it's not possible to provide a step-by-step solution for the expression provided. The student is advised to review and present the question accurately for further assistance.
Step-by-step explanation:
The provided student question is an expression that requires proof; however, due to the inclusion of unclear elements or potential typos, it is not possible to give an exact method to prove the expression as stated: 2²ᵐ . i cot⁻¹ (p) . pi/(p² - 1)².
The terms and concepts in this expression suggest it involves advanced mathematics principles such as exponents, trigonometry, and algebra, which could align with high school or college-level coursework covering topics in complex numbers, trigonometric functions, and algebraic manipulation.
Unfortunately, without clearer context or a more specific set of operations and relationships for the variables and functions involved, providing a step-by-step proof is not feasible. We advise the student to review the question and ensure that all variables and operations are presented accurately to enable a valid and complete response.