Proof of inequality from reduction formula

Question

I have got all of this question except for (iii). How do you prove both of these inequalities? Thanks image

Answer 1

On $(0,\frac{\pi}{4})$, you have $0 < \tan \theta < 1 \Rightarrow \tan^n \theta < \tan^{n-2} \theta \Rightarrow I_n < I_{n-2}$

From $I_n + I_{n-2} = \frac{1}{n-1}, I_n < I_{n-2}$ it follows immediately that

$$2I_n = I_n + I_{n} < I_n + I_{n-2} = \frac{1}{n-1}$$

Using the recursion for $n+2$ and $n$ you have $I_{n+2} + I_n = \frac{1}{n+1}, I_{n+2} < I_{n}$, it follows

$$2I_n = I_n + I_{n} > I_{n+2} + I_{n} = \frac{1}{n+1}$$