Question

prove that tangent of angle of repose will give the coefficient of static friction​

Answer 1

This is for the situation as in the video when there is no extra force involved !

Ok, so your μ = tanθ + M/(m cosθ) has the right sign and you have a set of observations of M as a function of θ . You can investigate if μ  depends on θ .

 (*). But if you want to show that μ = tan θ μ=tan⁡θ directly, you'll have to find a way to vary μ μ and work with M = 0.

 

(*) μ = tan θ μ=tan⁡θ doesn't mean that μ μ varies with θ θ; it means that the angle at which sliding is about to start has a tangent with a value that is equal to  μ.