$(B\wedge \sim D) \vee (\sim C \wedge D) \\ \equiv [B \vee (\sim C \wedge D)] \wedge [\sim D \vee ( \sim C \wedge D)] \\ \equiv [(B \vee \sim C) \wedge (B \vee D) \wedge ( \sim D \vee \sim C) \wedge (\sim D \vee D)] \\ \equiv (B \vee \sim C) \wedge (\sim D \vee \sim C) \wedge (B \vee D) \\ \equiv [ \sim C \vee ( B \wedge \sim D)] \wedge (B \vee D) \\ \equiv [\sim C \wedge (B \vee D)] \vee [( B \wedge \sim D) \wedge (B \vee D)] \\ \equiv [\sim C \wedge (B \vee D)] \vee [(\sim D \wedge B \wedge (B \vee D)] \\ \equiv [\sim C \wedge (B \vee D)] \vee [(\sim D \wedge B) \vee F \\ \equiv [\sim C \wedge (B \vee D)] \vee [(\sim D \wedge B) \vee (\sim D \wedge D)] \\ \equiv [\sim C \wedge (B \vee D)] \vee [\sim D \wedge (B \vee D)] \\ \equiv \small(B \vee D) \wedge (\sim C \vee \sim D)$