Proving logic propositions

I'm a little lost with this, can you guys provide full solutions?

The following propositions are tautologies, prove each one of them without truth table.

1) $\displaystyle p \Rightarrow (p \vee q).$

This is $\displaystyle \sim p \vee (p \vee q) \equiv ( \sim p \vee p) \vee q \equiv T \vee q \equiv T.$ (Where $\displaystyle T$ is a tautology.) Is this right? When can I apply $\displaystyle \sim p \vee (p \vee q) \equiv ( \sim p \vee p) \vee q$? Only when having all $\displaystyle \vee$ or $\displaystyle \wedge$?

2) $\displaystyle (p \Leftrightarrow q) \Leftrightarrow (p \wedge q) \vee ( \sim p \,\wedge \sim q).$ I dunno how to solve this one.

3) $\displaystyle \left[ {(p \Leftrightarrow q) \wedge (q \Leftrightarrow r)} \right] \Rightarrow (p \iff q).$ I dunno how to solve this one either.

4) $\displaystyle \left[ {(p \wedge \sim q) \Rightarrow \sim p} \right] \Rightarrow (p \Rightarrow q).$

This would be $\displaystyle \left[ { \sim (p \wedge \sim q) \vee \sim p} \right] \Rightarrow (p \Rightarrow q) \equiv ( \sim p \vee q \vee \sim p) \Rightarrow (p \Rightarrow q).$ How do I proceed there?

5) $\displaystyle \sim (p \Leftrightarrow q) \Leftrightarrow ( \sim p \Leftrightarrow q).$ Help here too.

6) $\displaystyle \left[ {(p \Rightarrow \sim q) \wedge ( \sim r \vee q) \wedge r} \right] \Rightarrow \sim p.$ Another help here.

7) $\displaystyle \left[ {p \wedge (p \Rightarrow q)} \right] \Rightarrow q.$

We have $\displaystyle \left[ {p \wedge ( \sim p \vee q)} \right] \Rightarrow q \equiv (p \wedge \sim p) \vee (p \wedge q) \Rightarrow q.$ What's next?

8) $\displaystyle (p \wedge q) \Leftrightarrow \left[ {(p \vee q) \wedge (p \Leftrightarrow q)} \right].$ Help here again :(

9) $\displaystyle (p \wedge q \Rightarrow r) \Leftrightarrow (p \wedge \sim r \Rightarrow \sim q).$ Need help here too.

That's it, I'm lost with this, I hope you can help me. :)