I_n=\left.-\sin^{n-1}x\cos x\right|_0^{\pi/2}+(n-1)\int_0^{\pi/2}\sin^{n-2}x\cos^2x\,dx=(n-1)\int_0^{\pi/2}\sin^{n-2}x(1-\sin^2x)\,dx=(n-1)(I_{n-2}-I_n)