Written by Colin+ in proof, trigonometry.

First up, a horrible confession: I like teaching the higher-lever core maths modules (C3 and C4), because they're closer to 'real' maths than the AS-level ones. One of the things that sets them apart is the introduction of proofs, usually for trigonometry1.

And a lot of students struggle with it -- it's not something you've seen before, it's an area you're not necessarily familiar with, and it's something that's so natural to most teachers of maths that it gets glossed over as trivial.

With that in mind, I wanted to give you some pro tips on making things easier.

OK, here's what an identity proof is: it's a list of expressions that are exactly the same as one side of the identity. Eventually, you come up with one that's exactly the same as the other side of the identity. For example: if you had to prove that $\sec^2(x) - \tan^2(x) \equiv 1$, you could start by saying:

"Well, I know $\sec^2(x)$ is the same as $\frac{1}{\cos^2(x)}$, and $\tan^2(x)$ is the same as $\frac{\sin^2(x)}{\cos^2(x)}$, so I can write something equivalent to the left hand side":

$\frac{1}{\cos^2(x)} - \frac{\sin^2(x)}{\cos^2(x)}$

"Oh! I have a pair of fractions with the same denominator. Let me combine them to get another thing that's the same":

$\frac{1 - \sin^2(x)}{\cos^2(x)}$

"That's looking simpler. I know the identity for linking $\sin^2(x)$ and $\cos^2(x)$, so I can replace that top with $\cos^2(x)$..."

$\frac{\cos^2(x)}{\cos^2(x)}$

"And that's clearly 1!"

You probably spotted a couple of things in there: I used the identities for $\sec(x)$ and $\tan(x)$ (you need to know that $\sec(x) \equiv \frac{1}{\cos(x)}$, $\cosec(x)\equiv \frac{1}{\sin(x)}$, $\tan(x) \equiv \frac{\sin(x)}{\cos(x)}$ and $\cot(x) \equiv \frac{1}{\tan(x)}$ or $\frac{\cos(x)}{\sin(x)}$; and I used the 'big daddy' identity:

$\sin^2(x) + \cos^2(x) \equiv 1$

You will use that one over and over again. Get used to it.

In other contexts, you're likely to need the formulas for $\sin(A+B)$ and its friends, which you'll find in your formula book.

It pays to keep one eye on the 'other side' of the proof so you can see where you're trying to end up. You're not allowed to move things from one side of an identity to the other (so, I couldn't have multiplied both sides by $\cos^2(x)$ in the example above because then I wouldn't have a list of equivalent expressions any more), but you can take hints from where you're headed.

Up there, I knew I needed to make the top and bottom of the fraction the same in order to end up with 1.

The nice thing about equivalent expressions is that they're equivalent both ways. While it's usually simplest to start from the complicated side and make it simpler, you can just as well complicate the simple side if that's what floats your boat.

For instance, it might be a bit easier to aim for $\frac{1}{\cos(x)}$ than for $\sec(x)$ -- they're equivalent, so it's ok to aim for either.

If there's one thing that messes people up with identity-type proofs, it's screwing up fractions. Be really careful about what you cancel, be really confident about doing arithmetic with fractions, and generally try to make them as simple as possible as quickly as possible.

You can whine 'but I don't *like* fractions!' as much as you like. Go ahead. I'll wait here. Did that change the syllabus? Let me check... nope, no, it didn't change the syllabus at all, you still need to be good at them. That was a waste of energy, wasn't it? You may as well have spent the time practising your fractions.

When I was at school, in the olden days, you wrote QED at the end of a proof. Nowadays, once you've shown something to be true, the done thing is to draw a little solid square. Trust me, it looks classy (and it shows you know what you're doing) ■

* Edited 2016-04-15 to fix LaTeX problems.

- There's sometimes a wee bit of proof in C1 and C2, but C3 is where it really kicks off. [↩]