The time taken for an object to slide down 45° rough inclined plane is n times the time taken for it to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is

From s = ut+$\frac{1}{2}a{t}^2$,$t=\sqrt{\frac{2s}{a}}$

For smooth plane a = g sin $\theta$

            For rough plane, a' = g(sin $\theta$ - $\mu$ cos $\theta$)

t' = nt$\Rightarrow$ $\sqrt{\frac{2s}{g(\sin \theta -\mu \cos \theta )}}=n\sqrt{\frac{2s}{g\sin \theta }}$

$n^2$g(sin $\theta$ -  $\mu$cos $\theta$) = g sin $\theta$

            When $\theta$ = 45°, sin $\theta$ = cos $\theta$= $1/\sqrt{2}$

Solving, we get $\mu =\left(1-\frac{1}{n^2}\right)$