"tangent of the angle of friction"
Do you mean the tangent of the angle in which an object starts to slide down an incline? If so
force down slide=mg*sinTheta
force up slide (friction)=mu*mg*cosTheta
when sliding starts, they are equal
mg*sinTheta=mg*mu*cosTheta
mu=TanTheta
show that, the tangent of the angle of friction is equal to the coeficient of friction.
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