orthonormal basis of given function

Hello all,

How can i write code for finding orthonormal basis functions for polynomial functions, if i dont want to use symbolic algebra.

for example,

vk = legendreP(n,x) - (legendreP(n,cos(t))*x)/cos(t), n = 2,3,4,5, .....

is my given function and i want to find its orthonormal basis function.

I tried to write this in symbolic form but post processing to its results took huge computational cost.