IB Math HL Trigonometry Test 4 2015 P2 Q5

Let \(T_n(x)=\cos(n\arccos x)\), where \(x\in[-1,1]\) and \(n\) is a positive integer.

1. Find \(T_1(x)\).[1]
2. Show that \(T_2(x)=2x^2-1\).[2]
3. [5]
   1. Simplify \(\cos(A+B)+\cos(A-B)\).
   2. Hence show that \(T_{n+1}(x)+T_{n-1}(x)=2xT_n(x)\).
4. Express \(T_3(x)\) as a cubic function of \(x\).[2]