simplify the expression using boolean algebra
(a+c).(a+a.d)+a.c+c
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(A + B) + (B + AA)(A + B) Original Expression
AA + AB + (B + A)A + (B + A)B Idempotent (AA to A), then Distributive, used twice.
AB + (B + A)A + (B + A)B Complement, then Identity. (Strictly speaking, we also used the Commutative Law for each of these applications.)
AB + BA + AA + BB + AB Distributive, two places.
AB + BA + A + AB Idempotent (for the A's), then Complement and Identity to remove BB.
AB + AB + AT + AB Commutative, Identity; setting up for the next step.
AB + A(B + T + B) Distributive.
AB + A Identity, twice (depending how you count it).
A + AB Commutative.
(A + A)(A + B) Distributive.
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