(i)Draw graph of y = (tan x) n , n ∈ 
N, x ∈ 
. Hence show
0 < (tan x) n+1 < (tan x) n , x ∈ 
y = (tan x) n , n ∈ 
N, x ∈ 
0 < (tan x) n+1 < (tan x) n , x ∈ 
(ii)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 that 1/(2n + 2) < A n < 1/(2n − 2).
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) 0 < tan x < 1, when 0 < x < π /4, we have
0 < (tanx) n + 1 < (tan x) n for each n ∈ N
(ii) we have A n = 
⇒ 
⇒ A n + 1 < A n
Now, for n > 0, A n + A n + 2 = 
= 
= 
= 
.
Similarly A n + A n–2 = 
since A n + 2 < A n + 1 < A n , we get A n + A n + 2 < 2A n
⇒ 
< 2A n ⇒ 
......... (1)
Also for n > 2, A n + A n < A n + A n – 2 = 
⇒ 2A n < 
......... (2)
⇒ A n < 
Combining (1) and (2) we get Hence Proved.
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