Question

Let U_n = ∫((1 – cos nx)/(1 – cos x)) dx for x ∈ [0,π], where n is a positive integer, show that U_n+2 + U_n = 2U_n+1. Hence, deduce that ∫(sin² nθ/sin² θ) dθ for θ ∈ [0,π/2] = nπ/2.  

Answer 1

We have, 

U_n+2 – U_n+1

Therefore, 

(U_n+2 – U_n+1) – (U_n+1 – U_n)

= 0 

Thus, U_n+1 + U_n = 2U_n+1

Now, 

(Let 2θ = t)