Final answer:
The student's question involves proving a trigonometric identity, however, there is not enough relevant information provided to directly solve this problem.
Step-by-step explanation:
The student asked to prove that if tan(1/2x)=cosec(x)-sin(x), then tan²(1/2x)=-2±√5. However, the information provided does not directly solve this problem and seems not relevant for the trigonometric identity provement. The typical approach to prove such trigonometric identities involves using trigonometric formulas and identities like the Pythagorean identities, sum and difference formulas, and double-angle formulas. To properly address the question, one would typically express the functions in terms of common trigonometric functions and manipulate the equation to show that the given identity holds true. It's important to check for any potential restrictions on the variable x, and solve the equation step by step accordingly.