are you sure it's not I_{n-1}? i checked it with mathematica and n-2 doesn't work, n-1 does.
I_n = \int_0^{\pi/4} \tan^{2n} x\,\text{d}x\\ = \int_0^{\pi/4} \tan^{2n-2}x (\sec^2 x - 1)\,\text{d}x \\ = \int_0^{\pi/4} \tan^{2n-2}x .\sec^2 x\,\text{d}x - \int_0^{\pi/4} \tan^{2(n-1)}x\,\text{d}x...