Prove that angle of friction is equal to angle of repose in the limiting friction

Prove that angle of friction is equal to angle of repose in the limiting friction.

Consider the figure .
image
The weight W can be resolved into two rectangular components. W cos f and W sin f. The component W cos f balances the normal reaction R while the component W sin f is equal to the limiting friction flimiting
image
image
image
∴ We notice that the angle of repose or sliding and the angle of friction are both equal.