I have written on different occasions about the importance of both empathy and resolution when the customer's expectations have not been met, and about the importance of an apology. When dealing with an angry customer, it's important to focus on resolving the issue at hand, to the best of your ability. However, if you don't show any empathy or acknowledgment for the customer's frustration, you still aren't going to provide an optimal customer experience.
I hear many CSRs say, "the customer tells me 'I don't want your apology!' so I never apologize." I have heard customers say this (though data shows it is far less seldom than usually reported), and whenever I hear the statement made it is almost always followed with something like: "I want my issue resolved!"
The problem is not that the customer wants no empathy, the problem is that the customer feels that all he/she has received is empathy and no one is fixing the problem. When the customer says, "I don't want your apology!' it should be a red flag telling you that empathy/resolution is out of balance and you need to focus on resolving the issue.
To avoid getting these out of balance, I recommend what I call a "empathy resolution statement." You offer a simple apology for the problem followed by a statement of what you are going to do to resolve it. Once done, you can usually move on to focusing on resolution knowing that you've adequately expressed empathy.
"I'm sorry that (describe the unmet expectation). What I can/will do is (describe what action you will take)."
Customer was supposed to receive a call back and didn't.
"I'm sorry we didn't call you back. What I will do is pull up your account and find out what I can do to resolve this for you."
Order didn't arrive.
"I apologize that you didn't get your order as expected. I can check the order for you and let you know what the tracking information tells us."
Customers called several times with the same issue and it hasn't been resolved.
"I'm sorry we've let you down on this issue. I will do everything in my power to get this resolved."
A simple empathy resolution statement will provide the customer with a balanced approach that leads to focus on resolution, but doesn't completely ignore the customer's need for empathy. As with all service skills, each CSR needs to find ways to make the statement conversational and incorporate a wording that is natural and comfortable. If both the empathy and resolution portions of the statement are clearly communicated, you will often set yourself up for a successful service experience.