relation between first kind Chebyshev poly and second kind Chebyshev poly

Question

How do you prove following relation between Chebyshev poly of first kind and Chebyshev poly of second kind: $$dT_n(x)/dx=nU_{n-1}(x)$$

Answer 1

Since $T_n(x) =\cos(n\arccos(x)) $ and $U_n(x) =\frac{\sin((n+1)\arccos(x))}{\sin(\arccos(x))} $,

$\begin{array}\\ T_n'(x) &=-(n\arccos(x)))'\sin(n\arccos(x))\\ &=-n\frac{-1}{\sqrt{1-x^2}}\sin(n\arccos(x))\\ &=\frac{n}{\sqrt{1-(\cos(\arccos(x))^2}}\sin(n\arccos(x)) \qquad\text{(this is a sort of sneaky part)}\\ &=\frac{n\sin(n\arccos(x))}{\sin(\arccos(x))}\\ &=nU_{n-1}(x) \end{array} $