Derivation of ripple factor equation for a full-wave rectifier with filter
The RMS value of a triangle wave is its peak value/sqrt(3).  In your case the peak value of the triangle wave is Vr/2, not Vr which is the peak-to-peak value. This correction will produce a factor of 4 in your final result instead of 2.  If you then make the approximation that RC>>T, then CRf>>1.  With that approximation, the denominator in your final equation can be approximated as 1 since 1/(4CRf)&lt;&lt;1.  The result is that your final equation reduces to the desired result.