Tautology propositions

Question

Anyone knows, considering $a$, $b$ and $c$ as propositions, if those propostions below are both tautologies? What I found out yet is that they are, but im not totally sure.
$((a∨b)∧((a→c)∨(b→c)))→c$
$a→((¬b→c)∨(¬b→¬c))$

Answer 1

The first statement is not necessarily true, consider when $a$ is false and $b\to c$ is false, in this case we may not have $c$.

The second one is true. Since the consequence $(¬b→c)∨(¬b→¬c)\equiv (b∨c)∨(b∨¬c)$ is always true.

Answer 2

The first one isn't even a valid statement. Suppose that $a$ is true and $c$ follows from $b$, but not necessarily from $a$. Then $c$ is not a guaranteed consequent from your premise even though the premise is true. However, you can change the second "or" to an "and," and your statement will be both valid and a tautology since it doesn't matter what your variables are.

Your second statement is a tautology. The consequent is always true, so no matter what $a$ is, the entire statement is true.