Question

Let A_n be the area bounded by the curve y = (tan x)ⁿ and the lines x = 0, y = 0 and x = π/4. Prove that for 

n > 2, A_n + A_n+2 = 1/(n + 1)

and deduce 1/(2n + 2) < A_n < 1/(2n - 2).

Answer 1

Since, A_n+2 < A_n+1 < A_n, We get

A_n + A_n+2 < 2 A_n