1. The problem statement, all variables and given/known data Let Tn(x)=cos(narccosx) where x is real and belongs to [-1,1] and n E Z+ Find T1(x). Show that T2(x)=2x^2 - 1. Show that Tn+1 (x) + Tn-1 (x) = 2xTn(x) Hence, prove by induction that Tn(x) is a polynomial of degree n. 3. The attempt at a solution Since cosθ=x and arccosx=θ we have: T1(x)=cos(arccosx)=cosθ=x. T2(x)=cos(2arccosx)=cos2θ=cos^2 θ - sin^2 θ = 2cos^2 - 1 = 2x^2 - 1. Tn+1 (x) + Tn-1(x) = cos((n+1)arccosx) + cos((n-1)arccosx) = 2cos(narccosx)cos(arccosx)=2Tn(x)*x. As for the induction we have: For n=1 T1(x)=x a polynomial of degree 1. Assuming for n to be true, i.e.Tn(x)=cos(narccosx) is a polynomial of degree n, prove T(n+1)(x)=cos((n+1)arccosx) is of degree n+1. So, cos((n+1)arccosx)=cos(nθ + θ)=cosnθ*x - sinnθsinx ... Obviously this is not the way to approach the problem. Anyway, I would prefer to hear tips from you rather than a complete solution. Thanks in advance.