The tangent and cotangent versions, and the angles they refuse to cover

0:00There are four more identities in this family, and they are the ones that arrive with small print. The tangent of a sum, the tangent of a difference, and the same two for the cotangent. Each one comes with a line in front of it naming the angles it does not cover. Most people read that line once, file it under legal boilerplate, and never look at it again. It is not boilerplate.

0:28It is the proof, left out in the open where you can see it. Every one of these four is made by dividing, and there is exactly one thing in arithmetic you are not allowed to divide by. The angles a condition rules out are the angles where you would have divided by nothing. So if you can point at the division a condition is protecting, you never have to remember the condition at all.

0:55And if you cannot point at it, you have not seen the proof. Start with what the two quotients are. The tangent is the second coordinate over the first. The cotangent is the first over the second. That is the whole definition, and it is the only way either of them will enter this. Now the move, which is the same move every time. Take an expansion you already have, put it over another expansion you already have, and tidy the fraction by dividing the top and the bottom by one carefully chosen product.

1:34There are only two decisions in the whole topic: which of the two quotients you want, and whether the second angle arrives as itself or negated. Two decisions, two ways each, four results. No new geometry appears anywhere, and nothing here gets proved twice. Take the tangent of A plus B. By definition it is the second coordinate of A plus B, over the first coordinate of A plus B.

2:06Both of those we already have as expansions. The top is the sine of A times the cosine of B, plus the cosine of A times the sine of B. The bottom is the cosine of A times the cosine of B, minus the sine of A times the sine of B. Both of those are checked here at all five hundred and seventy six pairs before anything is done to them.

2:33That fraction is already true, and it is already useless: four products, and not a tangent in sight. Before we tidy it, kill the guess. You might hope the tangent of a sum is just the sum of the two tangents. Over the four hundred and forty four pairs where all three tangents exist, that guess is right at one hundred and twenty four of them, and wrong at three hundred and twenty.

3:01Right more than a quarter of the time is exactly how a wrong rule survives. So tidy it, and watch what the tidying costs. Divide the top and the bottom by the product of the two cosines. The top becomes the tangent of A plus the tangent of B. The bottom becomes one, minus their product. There is the identity, and it took one division. At four hundred and eighty four of the five hundred and seventy six pairs that divisor is something you are allowed to divide by, and at all four hundred and eighty four of those the top and the bottom come out in exactly that shape.

3:46Now look at the bottom again. One minus the product of the two tangents is nothing exactly when the first coordinate of A plus B is nothing. Nobody put that there. The third condition arrived on its own, out of the arithmetic, the moment the fraction was written down. Three angles get ruled out, and they are doing three different jobs. Ruling out A keeps the tangent of A in existence, and keeps half of the divisor from being nothing.

4:21Ruling out B does the same for the other half. Ruling out A plus B keeps the left hand side in existence. It is tempting to think these are three ways of saying one thing, and they are not. Across the five hundred and seventy six pairs there are forty where A is the only one of the three that fails. Forty where B is the only one. And forty where A plus B is the only one.

4:49Each of the three conditions is the only thing standing between you and a false statement, at forty pairs of its own. A condition on the combined angle does not imply the conditions on the two angles that made it, and there are eighty four pairs to prove it. Here is the case worth carrying around for the rest of your life. Let A be a quarter turn, and let B be an eighth of a turn.

5:18A minus B is an eighth of a turn, and the tangent there is one. The left hand side of the difference identity is a perfectly ordinary number. The right hand side wants the tangent of A, and A is a quarter turn, where there is no tangent at all. A live left side and a dead right side, in the same line. So an identity does not only fail where its left hand side fails.

5:47And this is not a freak: at eighty four of the five hundred and seventy six pairs the left side is a number and the right side is not. Going the other way, the count is zero. Not once does the right hand side offer a value where the left hand side has none, which is the one direction that would make the identity a lie rather than a gap.

6:13The difference form is not proved again. Put minus B where B stood. The tangent of a negated angle is the negative of the tangent, at all twenty two inputs where it exists. So the sum in the top becomes a difference, and the minus in the bottom becomes a plus, and nothing else moves at all. The identity is new. The conditions are not. Whatever the substitution walks in with, it keeps: four hundred and forty four pairs where both sides are numbers, eighty four where only the left one is, and zero the other way.

6:54The same three counts, because it is the same divisor. A difference form written with no condition beside it is not a difference form that has no condition. Now the cotangent, and it is the same move with one thing changed. The cotangent is the first coordinate over the second, so the two expansions go the other way up. The cosine expansion is now the top and the sine expansion is now the bottom.

7:26And the tidying divides by the product of the two sines instead. The top comes out as the product of the two cotangents, less one. The bottom comes out as the two cotangents added. Four hundred and eighty four of four hundred and eighty four again, so the shape is not a coincidence of one example. Same move, different divisor. And a different divisor is a different list of angles you may not hand it.

7:58That is worth making precise, because the two lists are usually remembered as one. Of the twenty four inputs, the tangent has no value at two of them: the odd multiples of a quarter turn, where the first coordinate is nothing. The cotangent has no value at two as well: the whole multiples of a half turn, where the second coordinate is nothing. Those two lists have zero members in common.

8:28Not mostly different. Nothing in common at all. Counted as pairs, the tangent identity leaves out one hundred and thirty two of the five hundred and seventy six, the cotangent identity leaves out one hundred and thirty six, and only twelve pairs are left out by both. Three hundred and twenty pairs are covered by both of them. So there is no such thing as the condition on these identities.

8:58There are two conditions, they guard two different divisors, and they overlap almost nowhere. That does not mean four conditions have to be remembered. It means two. The sum and the difference can be said once, with a plus or minus in the middle. For both tangent forms: none of A, none of B, and none of A plus or minus B may be an odd multiple of a quarter turn.

9:28For both cotangent forms: none of the three may be a whole multiple of a half turn. Two lines covering four identities. And the plus or minus is not a piece of shorthand. It is the accurate statement, because the negation that produced the difference form carried the condition across with it, and a condition that is inherited is still a condition. The division pays a second time, in something that looks unrelated.

10:00Both coordinates take a whole turn to come back to themselves. Twenty four steps, and not one step fewer. Their quotient comes back after twelve. Half a turn reverses both coordinates, so the two minus signs cancel and the quotient is unchanged. The tangent repeats twice as often as the numbers it is built from, and that is the reason, not a separate fact to be learnt beside it. Which means a half turn can be peeled off any angle before you start.

10:34Take thirteen steps round, which is a half turn and one step. Peel off the half turn and one step is left. One step is forty five degrees less thirty. The difference identity, with the tangents one and one over root three, gives two minus root three. An awkward angle, and nothing about it was awkward. The move is worth far more than the four results it was invented for.

11:03Take the sine of a sum, over the sine of a difference. Expand both, divide the top and the bottom by the same product of cosines, and out comes the sum of the two tangents over their difference. That holds at four hundred and forty of the four hundred and forty pairs where both sides are numbers, and it was never one of the four results. Take the tangents at an angle, its double and its triple.

11:33Their product equals the triple, less the double, less the single, at all fourteen inputs where the three of them exist together. And one caution, because it is the one that costs marks. The three cotangents at an angle, its double and its triple, combined with the signs the derivation actually gives, come to exactly one at all sixteen inputs where they exist. Add the same three products instead of alternating them, and it is one at only four of the sixteen.

12:08A sign copied down carelessly is a different statement, and it is right a quarter of the time, which is precisely how it survives. So here is the whole of it. Two quotients, two divisors, one tidying move, four results. The conditions were never attached to the results afterwards. They came out of them. Every angle that is ruled out is a place where a division would have had nothing to divide by, or a quotient that does not exist was being asked for.

12:44That is why the tangent forms rule out odd quarter turns and the cotangent forms rule out half turns, and why the two lists share nothing. If you can point at the divisor, you can write the condition down without ever having memorised it. And if you cannot, then what you kept was the by-product, and what you threw away was the argument.