Looks like you found the correct answer, only
Therefore,
Find the vectors T, N, and B at the given point.
r(t) = {cos(t), sin(t), ln(cos(t))}, (1,0,0)
I got T(t)= r'(t)/|r'(t)|= (-sint, cost, -sint/cost)/square root(1+tan^2t) = (-sint,cost,-tant)=(0,1,0) because t=0.
then for N(t)= T'(t)/|T'(t)|=(-cost,-sint,-sec^2)/ sqaure root(1+sec^4t)=(-cost/sqaure root 2,-sint/sqaure root 2, sec^2t/sqaure root 2)
I am just wondering if my math is correct or if I went wrong somewhere.
B(t)= T(t)xN(t)= -1/sqaure root2[(sect+sinttant)i+(sint(sec^2t + 1)j -k]
=(-1/sqaure root2,0,1/sqaure root 2)
Did I solve this correctly?