c
Given, \(\theta=45^{\circ}, s_{1}=s_{2}, u=0\)

On the rough incline, \(a_{1}=g(\sin \theta-\mu \cos \theta)\)

\(t_{1}=\) time taken

On the frictionless incline, \(a_{2}=g \sin \theta\)

\(t_{2}=\) time taken, \(t_{1}=2 t_{2}\)

\(s=u t+\frac{1}{2} a t^{2}\)

\(s_{1}=0+\frac{1}{2} g(\sin \theta-\mu \cos \theta) t_{1}^{2}\)

\(s_{2}=0+\frac{1}{2} g \sin \theta t_{2}^{2}\)

As \(s_{1}=s_{2}\)

\(\frac{1}{2} g(\sin \theta-\mu \cos \theta) t_{1}^{2}=\frac{1}{2} g \sin \theta t_{2}^{2}\)

\(\frac{\sin \theta-\mu \cos \theta}{\sin \theta}=\frac{t_{2}^{2}}{t_{1}^{2}}\)

\(\Rightarrow 1-\mu \cot \theta=\frac{t_{2}^{2}}{\left(2 t_{2}\right)^{2}}\)

\(\Rightarrow 1-\mu \cot \theta=\frac{1}{4}\)

\(\Rightarrow \mu \cot \theta=1-\frac{1}{4}=\frac{3}{4}\)

\(\therefore \mu=\frac{3}{4 \cot \theta}\)