Trading a sum of two ratios for a product, and back again

0:00Here is a sum of two sines. Now try to do anything with it. You cannot cancel a piece of it, because there is no factor to cancel. You cannot divide part of it away. You cannot even simplify it, because there is nothing sitting outside a bracket. A sum is algebraically inert. A product is the opposite: it has factors, and factors lift out and cancel. So the whole of this topic is one trade — turn the sum into a product, and suddenly there is something to do.

0:36But before the trade, kill the guess. The sum of two sines is not the sine of the sum. Two quarter turns have second coordinates of one each, adding to two, while a half turn's second coordinate is nothing at all. And across five hundred and seventy six pairs of angles, that guess is accidentally right at seventy of them, which is exactly how a wrong rule survives being tried once.

1:06The trade costs no new geometry. Take the two expansions you already have: the first coordinate of a sum, and the first coordinate of a difference. They are the same two products, and the only thing that changes between them is one sign. Both hold at all five hundred and seventy six pairs. Now add them. The two second-coordinate products carry opposite signs, so they kill each other, and what is left is twice a product of two first coordinates.

1:41Five hundred and seventy six of five hundred and seventy six. Now subtract them instead. This time the first-coordinate products cancel and the other pair doubles, leaving minus twice a product of two second coordinates. Also five hundred and seventy six of five hundred and seventy six. Look at that minus. It is not decoration, it came out of the subtraction, and it is the one people drop most often.

2:15Do it again with the other pair. The second coordinate of a sum, and the second coordinate of a difference. Add them, and the terms that carry the second angle's second coordinate cancel, leaving twice a second coordinate times a first. Subtract them, and the other pair survives instead, leaving twice a first coordinate times a second. Both at five hundred and seventy six of five hundred and seventy six.

2:46Four equations. No new geometry, no new picture, and nothing proved twice. They are the four expansions, added and subtracted in pairs, and the entire topic is already sitting in them. Except that as they stand, you cannot use a single one of them. Look at what each equation is written in terms of. The right-hand sides are products at the two angles you started with. The left-hand sides are about their sum and their difference.

3:20Now think about the situation you are actually in. Somebody hands you a sum of two cosines. You are holding the two angles on the LEFT. You do not have the two angles that generated them, and nothing in the equation tells you what they were. That mismatch is the whole problem, and it is the only reason the next step exists. The step is not deep, and it is not a trick.

3:47It is renaming, done carefully. Give the two angles you are actually holding their own names. Then ask what the original pair must have been. One of them was the total of your two, halved. The other was the gap between your two, halved. Call those the half-sum and the half-difference, and the four equations are suddenly written in things you have. That is the whole change of variable. But notice what it cost, because it is a real cost and it never goes away.

4:22Going this direction, you have to halve. Going the other direction — starting from a product and wanting a sum — you never halve anything, because the two angles of the answer are just the total and the gap. Measured on a circle of twenty four marked angles: the product direction lands on a marked angle every time, five hundred and seventy six of five hundred and seventy six. The sum direction has to halve, and the halved angle is a marked one for two hundred and eighty eight of the five hundred and seventy six.

5:00That is a fact about the twenty four marks, not about the identity. But the halving is real in both. So here are the four, in the form you will actually use them. A sum of two first coordinates is twice the first coordinate at the half-sum, times the first coordinate at the half-difference. A difference of two first coordinates is MINUS twice the second coordinate at the half-sum, times the second coordinate at the half-difference.

5:33A sum of two second coordinates is twice the second at the half-sum times the first at the half-difference. And a difference of two second coordinates is twice the first at the half-sum times the second at the half-difference. Exactly one of those four carries a minus sign out in front. One of four. And here is why it gets dropped: over the five hundred and seventy six pairs, the version without the minus is still right at ninety two of them.

6:07Not right often enough to trust, and right far too often to notice. The two out in front is load-bearing as well. Drop it, and what is left is wrong at every pair except ninety two. Now here is something worth more than either warning. Suppose you drop the half-difference factor entirely and keep just twice the first coordinate at the half-sum. That version is right at seventy of the five hundred and seventy six, so it is wrong almost everywhere.

6:41And yet: test it on a pair of EQUAL angles, and it passes at all twenty four. Every single one. Because when the two angles are equal, the half-difference is nothing, its first coordinate is one, and a factor of one is invisible. Test the same broken version with the SECOND angle set to nothing instead, and it fails at twenty one of the twenty four. Same formula, same fault, and one test sees none of it while the other sees almost all of it.

7:15Choosing the test is a skill, and equal angles is almost always the wrong choice. You should not be memorising four formulas. There are two sentences underneath them. When the two terms are ADDED: the half-difference takes a first coordinate, and the half-sum keeps whichever coordinate you started with. When they are SUBTRACTED: the half-difference takes a second coordinate, and the half-sum switches to the other one. That is it.

7:48Two sentences, and they are checked here against the circle at every pair. Three of the four parts come out exactly right, five hundred and seventy six of five hundred and seventy six each. The fourth comes out exactly right as well, once you put its minus back. So the sign is the one thing the pattern does not tell you, and it is the one thing worth remembering separately.

8:16Everything else you can rebuild from a sentence. Now run the four backwards. Twice a product of two first coordinates is a sum of two first coordinates. Minus twice a product of two second coordinates is a difference of two first coordinates. And so on, all four. It looks like four more results to learn. It is not. It is the same four equations with the two sides written the other way round, and here that is a measurement rather than a remark: going the long way — forming the product's two angles, then hunting backwards for the pair that produces them — lands on the same equation, at all two thousand three hundred and four.

9:06Two more things worth noticing in the four. The last two parts are each other with the two angles exchanged, at all five hundred and seventy six. The first two do not even notice the exchange. So four parts, two sentences, and really one idea. Now watch what this is for. Take the sum of the first coordinates at seven times an angle and at five times it, over the difference of the second coordinates at the same two.

9:39As it stands there is nothing to do with it. Convert the top and convert the bottom. Both have the same half-sum, six times the angle, and the same half-difference, the angle itself. The top becomes twice a product with a first coordinate at six times the angle in it. The bottom becomes twice a product with the SAME first coordinate at six times the angle in it. It cancels, and what survives is a first coordinate over a second — the cotangent.

10:13Counted over the twenty four inputs: both sides are numbers at ten of them and agree at all ten, the right side alone is a number at twelve, neither is at two, and there is no input where the left lives and the right does not. And the factor you cancelled is nothing at twelve of the twenty four, which is exactly where the left side stopped existing. The cancellation and the hole are the same event.

10:43Three terms look like they cannot be converted, because the rule takes two. Pair the outer two, and the factor that comes out is usually shared with the middle one. Here is the shape. First coordinates at four, three and two times an angle, over second coordinates at the same three. On top, the outer pair gives twice a first coordinate at three times the angle, times a first coordinate at the angle.

11:13The middle term is a first coordinate at three times the angle. So the whole top is that, times a bracket, at all twenty four. The bottom does the same with a second coordinate at three times the angle, times the SAME bracket, at all twenty four. The brackets cancel and the cotangent at three times the angle is left. One honest caution: that bracket is nothing at two of the twenty four inputs, and there you are not allowed to cancel it.

11:46Both sides are numbers at eighteen inputs and agree at all eighteen. Turn the quotient upside down and still call the answer a cotangent, and you are right at twelve of them — because upside down it is the tangent, and the tangent and the cotangent agree half the time. Now the one that breaks the habit. Not every problem wants a sum turned into a product. Take a difference of two products of first coordinates, and you are asked to show it is a product of two second coordinates.

12:23The obvious move is to convert the answer. The move that works is to go the other way first. Run each product backwards into a sum, and four terms land on the board. One pair of them strikes out — and not by luck: the second product's second angle is exactly minus the first product's, so the two terms carry the same reading with opposite signs, at all twenty four inputs.

12:52Four terms out, two gone, two left. And the two that are left are a difference of first coordinates, which is precisely what the sum direction converts, straight into a product of two second coordinates. Twenty four of twenty four. Both directions, in one problem, in that order. So the question is never which direction is correct. It is which direction produces something you can cancel. Two shapes come up often enough to keep.

13:26The first: a difference of two squared second coordinates is the second coordinate at the sum, times the second coordinate at the difference. Five hundred and seventy six of five hundred and seventy six. That is just two of the four parts multiplied together, and it turns a whole family of questions into one line. The second: four terms rather than three. Second coordinates at one, three, five and seven times an angle.

13:58Pair the outermost two and pair the innermost two, and BOTH pairings hand back a factor at four times the angle, at all twenty four. That shared factor is the whole answer. And one that closes a loop: take two points on the circle and square the gap between their first coordinates, square the gap between their second coordinates, and add. That comes to four times a squared second coordinate at the half-difference, at every pair — and it also comes to two, less twice the first coordinate at the gap between them.

14:36Which is the chord computation these identities were built on in the first place. So, faced with something to prove, what do you actually ask? Not which formula. Ask where a common factor could come from. If the top and the bottom are both sums, convert both — the half-sums usually match, and the shared factor cancels. If there are three terms, pair the outer two and look for the middle one in what comes out.

15:08If there are products already, run them backwards and see what cancels before you collect anything. And if a factor you want to cancel can be nothing, say so, because a cancellation is a division. Underneath all of it there is one move done four times: two expansions, added and subtracted, and then renamed. A sum has nothing to cancel. A product is nothing but things to cancel. That is the entire reason these exist.