Roots of the Chebyshev polynomials of the second kind. It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots of $U_n{(x)}+U_{n-1}(x)$ (by looking at the law ranks of $n$) also lies in $(-1,1)$, I also noticed that  for $(1-x)U_n{(x)}+U_{n-1}(x)$ the roots lie in  $(-2,2)$. But I don't have any idea how to prove that in general, I wonder, first, if these claims are true? and how can I start proving them?

See my answer here by writing chebyshev polynomial explicitly in terms of $x = \cos t$. 
Update: I also show there that the roots are all real.
