I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\theta [\sin\theta \cos(n-2)\theta + \sin(n-2)\theta \cos\theta] = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\theta \sin\theta \cos(n-2)\theta + \cos^2\theta \sin\theta \cos(n-3)\theta +... = $$ $$ = \sin\theta [\cos(n-1)\theta + \cos\theta \cos(n-2)\theta + \cos^2\theta \cos(n-3)\theta + \cos^3\theta \cos(n-4)\theta+...] = $$ $$ = \sin\theta [T_{n-1} + T_{1}T_{n-2} + T_{1}^2 T_{n-3} + T_{1}^3T_{n-4}+ ... ] $$ So $$ T_n'(x) = [T_{n-1} + T_{1}T_{n-2} + T_{1}^2 T_{n-3} + T_{1}^3T_{n-4}+ ... = \sum \limits _{i=0} ^{n-1} T_1^i T_{n-1-i} $$ I tried to use fact that $$ T_r(x)T_s(x) = 0.5 (T_{r+s}(x)+T_{r-s}(x)),$$ but I can't prove that $ T'_n(x) = \sum \limits _{i=0} ^{n-1} a_{ik}T_i(x),$ because I can't find $a_{ik}.$
Note that \begin{align*} \frac{\sin n\theta}{\sin\theta} &=\frac{e^{in\theta}-e^{-in\theta}}{e^{i\theta}-e^{-i\theta}}\\ &=\frac{(e^{i\theta}-e^{-i\theta})[e^{i(n-1)\theta}+e^{i(n-3)\theta}+\dots+e^{-i(n-1)\theta}]}{e^{i\theta}-e^{-i\theta}}\\ &=e^{i(n-1)\theta}+e^{i(n-3)\theta}+\dots+e^{-i(n-1)\theta}\\ &=\begin{cases} 2\cos((n-1)\theta)+2\cos((n-3)\theta)+\dots+2\cos(\theta) & n\text{ even}\\ 2\cos((n-1)\theta)+2\cos((n-3)\theta)+\dots+1 & n\text{ odd} \end{cases} \end{align*} So \begin{align*} T_n'(x)&= 2T_{n-1}(x)+2T_{n-3}(x)+\dots+\begin{cases}2T_1(x) & n\text{ even}\\T_0(x) & n\text{ odd}\end{cases}\\ &=2\sum_{r=0}^{\lfloor (n-1)/2\rfloor}T_{n-1-2r}(x)+\text{constant term correction} \end{align*} and similarly: \begin{align*} T_n''(x) &=2\sum_{r=0}^{\lfloor (n-1)/2\rfloor}T_{n-1-2r}'(x)\\ &=4\sum_{r=0}^{\lfloor (n-2)/2\rfloor}(r+1)T_{n-2-2r}(x)+\text{constant term correction}\\ T_n^{(3)}(x) & = 8\sum_{r=0}^{\lfloor (n-3)/2\rfloor}\binom{r+2}{2}T_{n-3-2r}(x)+\text{constant term correction}\\ T_n^{(4)}(x) & = 16\sum_{r=0}^{\lfloor (n-4)/2\rfloor}\binom{r+3}{3}T_{n-4-2r}(x)+\text{constant term correction}\\ \end{align*} etc.