(This is a very, very long answer because I want to highlight both the likely mindset of such a pupil, and possible approaches to win them over. Also, please check Student Poisoned Experience with Math in case it may help you.)
Understanding the Mindset
Another example could be an expression with non-round numbers, where I suggested to hide them behind α and its variations (esp. since such substitution made the necessary formulas and reductions easily visible). Nevertheless, he tirelessly rewrote them all the way through calculations and clung to their concreteness like to some kind of lifeline.
You sound like you're teaching a younger me. I feel sorry for you.
This was exactly what I did. I would reduce the problem, not to its simplest abstract form from which a solution becomes more apparent, but to the form conceptually most recognisable to me, which would lead me around and often down a blind alley. And when you would say that there is an easier way to do it, and you'd even show it, I'd just ignore it because I didn't understand what you were saying, and I wanted to get there on my own steam, dammit.
In short, you have a pupil who is disenchanted, depressed, and close to belligerent.
I feel that, as tutors, we tend to make a mistake. We see someone take the long way around and we offer a short-cut. We approach the subject from the perspective of a master, with insights of simplicity and foresight. By imparting our knowledge, we're hoping something will stick, and that their performance will improve.
The perspective of our pupils, however, is that of a frog trying to cross a busy highway: at some point, they just make the plunge, race screaming and flailing for the other side, and hope not to get hit by a truck. They can't see the short-cut because they don't understand the short-cut, even when it's pointed out with neon signs.
And that's the really hardest part here, I think: debugging the misconceptions op your pupil. Stepping out of our frame of reference, and getting into theirs.
"I don't need to understand it" and "I would like to do it without thinking". (*)
How's that working out for you, partner?
Help him out with the rules by putting them on display. For trigonometry, show the unit circle, and put an example in each quadrant. It'll help him visualize and internalize the relations between the trigonometric functions. At some point, remove the unit circle--but let him draw it himself, if he so chooses.
It will also help him realise that these 'rules' are not axiomatic, independent laws, but properties and relations; tools rather than obstacles.
(*) Objects follow the path of least physical resistance. People follow the path of least intellectual resistance.
Changing the Approach
How can I change such a student's approach to math?
Sneak the math onto him. Really. Seriously.
My brother had the same problem tutoring me in physics. I was horrible. I didn't get anything right. I hated it. I wanted to burn my course book after smothering it in the blood of a sacrificial goat.
We had been covering potential and kinetic energy, and then he decided we needed a better exercise. So we were going to calculate the horsepower of throwing a horse off the Niagara falls. Because the exercise was so far removed from what I personally experienced as the boring, horrible physics, I did not have the instant mindblock I would otherwise have (*). Suddenly, physics was a means to an end, and I had the (admittedly morbid) drive to find the answer.
A logic and philosophy professor at our university had this approach to teaching where he would start with anecdotes and slowly, sneakily slide into his courses. By the time students 'caught on' that he was back to teaching, he brought up another anecdote to reel them in.
Our probability and statistics professor would take topics from recent news for exercises and show either why the news report was highly misleading, or what the implications of something would be: error margins on polling, probabilities on game shows, casino games, poker, stuff like that.
(*) We didn't actually throw a horse off a cliff. My brother wasn't too big on empirical physics.
For a change I wanted to show him something engaging, e.g. how math allows us to build nice things using 3D animation software, but he would not make a connection. It was just a few cool pics useful only as long as they can be used to raise his status (e.g. his friends think them awesome).
3D animations can be cool and flashy, but they don't immediately, intuitively link back to math for a layperson.
Stick closer to home. Does he like video games? Maybe he played Call of Duty, and heard of the Coriolis effect. Or maybe he played Realm of the Mad God; that uses Voronoi diagrams for world generation. Maybe he's addicted to his cell- or smartphone; introduce satellite orbits or talk about GPS systems. Math itself is difficult to connect directly, so you may have to go through physics.
You're right: you can't make him go from loathe to love in an instant. Even now, I don't love mathematics, but I am no longer terrified of it, and I regularly use it as a powerful tool of abstraction.