In the Previous Problem, If We Also Require That $\Theta$ Be Bounded Everywhere, the Solution of the Eigenvalue Problem Is

In the previous problem, if we also require that $\Theta$ be bounded everywhere, the solution of the eigenvalue problem is

A) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \sin \theta ) , n = 1,2,3 , \ldots$
B) $\lambda = n ( n - 1 ) , \Theta = \sin ( n \theta ) , n = 1,2,3 , \ldots$
C) $\lambda = n ( n - 1 ) , \Theta = \cos ( n \theta ) , n = 1,2,3 , \ldots$
D) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \cos \theta ) , n = 1,2,3 , \ldots$
E) $\lambda = n ( n - 1 ) , \Theta = P _ { n } ( \sin \theta ) , n = 1,2,3 , \ldots$