I know this feeling intimately, having tried a little consulting and having a serious ethical bend towards the truth.

The fact is, the better I get at programming (and math and science) the more I know I don't know, and the less confident I am about any given problem. The thing I make myself remember though, is that I'm damn good at solving problems. I may not have implemented this algorithm yet, or used this technology stack or programming language, but rest assured in a couple hours I will know what I should look out for and how I should proceed.

When I speak with the confidence in my ability to learn, something I have proven to myself, I no longer feel like I am faking or deluding some innocent luddite. I am giving comfort to somebody who needs my problem solving skills.

At least, that's how I justify it :)