It is given that

\(I_n=\int_0^{\frac14\pi}\frac{\sin^{2n}x}{\cos x}\,dx,\qquad n\geq0.\)

Show that

\(I_n-I_{n+1}=\frac{2^{-\left(n+\frac12\right)}}{2n+1}.\)

Hence show that

\(\int_0^{\frac14\pi}\frac{\sin^6x}{\cos x}\,dx=\ln(1+\sqrt2)-\frac{73\sqrt2}{120}.\)