What would be evidence that President Kennedy used appeals in his inauguration address? I understand the pathos of his address just not the ethos or logos.
You are correct in seeing this as an address primarily given through "pathos." It certainly appeals to the emotional side of the United States citizens. However, a closer look will illustrate that both ethos and logos were on Kennedy's side as well.
First, to confirm, "ethos is appeal based on the character of the speaker." Someone who is able to appeal to the ethos of an audience can do so because of their character. President Kennedy was a senator before he was president and he had won the election, which means that he had won the trust of the majority of the American people.
Let's look at a section of his speech:
To those new states whom we welcome to the ranks of the free, we pledge our word that one form of colonial control shall not have passed away merely to be replaced by a far more iron tyranny. We shall not always expect to find them supporting our view. But we shall always hope to find them strongly supporting their own freedom -- and to remember that, in the past, those who foolishly sought power by riding the back of the tiger ended up inside.
As the leader of the free world, Kennedy is able to appeal "to those new states whom we welcome to the ranks of the free." He's in charge of those ranks, and thus has the authority for welcome them. Logically, we take the last line of the paragraph: "and to remember that, in the past, those who foolishly sought power by riding the back of the tiger ended up inside." Reason tells us that if you try to ride on the back of a tiger, you will be attacked and eaten. Here, he is saying that people who have newly realized freedom, must continue to fight to develop it, but that the United States will be there to help if necessary.
While logos and pathos are evident here in this paragraph, the speech appeals to ethos throughout.