The XNOR circuit can also be made with INVERTERS, NAND, and NOR gates just like the XOR. In fact, the circuits are almost identical.

     ___________________
      _______   =======
 C = ((A * B) * (A + B))    [given formula]

     ___________________
      _______
 C = ((A * B) * (A + B))    [A = A  {double negative}]

     =======   _______                _______   _   _
 C = (A * B) + (A + B)      [DeMorgan (A * B) = A + B]

               _______       =
 C = (A * B) + (A + B)      [A = A {double negative}]
                _   _                 _______   _   _
 C = (A * B) + (A * B)      [DeMorgan (A + B) = A * B]

This formula simplifies much easier then the XOR one did. For the XNOR function to be true, both A and B must be false or both A and B must be true. For the XOR function, these conditions are when the output is false. Literally, the XNOR is simply an inverted XOR.

      _________________
           _         _
  C = (A * B) + (B * A)     [given formula, XNOR is an
                                          inverted XOR]
      _______   _______
           _         _                 _______   _   _
  C = (A * B) * (B * A)      [DeMorgan (A + B) = A * B]
       _         _                     _______   _   _
  C = (A + B) * (B + A)      [DeMorgan (A * B) = A + B]
        _        _      _
  C = ((A + B) * B) + ((A + B) * A)
                      [A * (B + C) = (A * B) + (A * C)]
        _   _     _           _
  C = ((B * A) + (B * B)) + ((A * A) + (A * B))
                      [A * (B + C) = (A * B) + (A * C)]
        _   _                                    _
  C = ((B * A) + 0) + ((A * B) + 0)         [A * A = 0]
       _   _
  C = (B * A) + (A * B)                     [0 + A = A]

This alternative formula simplifies into the same equation. Remember the pass gate version of the XNOR gate.

With pass gates, each path can be seen in the circuit. When B is true, the bottom set of pass gates are on. The signal passed is A. However, when B is false the top set passes inverted A.