Let $n$ be an integer with $n \geqslant 1$. Are the following statements true or false? Give proofs.

(i) There exists a real polynomial $T_{n}$ of degree $n$ such that

$$T_{n}(\cos t)=\cos n t$$

for all real $t$.

(ii) There exists a real polynomial $R_{n}$ of degree $n$ such that

$$R_{n}(\cosh t)=\cosh n t$$

for all real $t$.

(iii) There exists a real polynomial $S_{n}$ of degree $n$ such that

$$S_{n}(\cos t)=\sin n t$$

for all real $t$.