(Solved): Problem 4: We define a sequence of polynomials (Tn(x))n=0 by T0(x)=1,T1(x)=x and Tn ...

Problem 4: We define a sequence of polynomials $${\left(T_n(x)\right)}_{n=0}^{?}$$ by $$T_0(x)=1,T_1(x)=x$$ and $$T_n(x)=2x{T}_{n?1}(x)?T_{n?2}(x),n?2.$$ So $$T_2(x)=2x^2?1,T_3(x)=4x^3?3x$$, etc. Show that $$T_n(\cos (t))=\cos (nt)$$ for $$n?0$$. [HinT: The trigonometric formula $$\cos (x)\cos (y)=\frac{\cos (x+y)+\cos (x?y)}{2}$$ may come in handy, with $$x$$ and $$y$$ equal to suitable multiples of $$t$$. We will consider it known, so you do not have to prove this formula before using it.]