Chebyshev Proof

[Guest]

Hi all,

Struggling with the below (not a clue how to prove it, could anyone help?):

Let [tex]T_{k }[/tex](x)=cos(k arccos x). Show that:

[tex]T_{n }(T_{m }(x))=T_{mn }(x)[/tex]

Any help would be appreciated.

[shyamjayakannan]

[tex]T_n(T_m(x))=T_n(\cos(m\cos^{-1}x))=\cos(n\cos^{-1}(\cos(m\cos^{-1}x)))=\cos(nm\cos^{-1}x)=T_{mn}(x)[/tex]