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

What to assume they know

• One distance calculation on the unit circle yields the cosine of a difference — results 3 and 4, the cosine expansions
• Shifts by a quarter, a half and a whole turn all fall out of the same two results — results 7 and 8, the sine expansions
• Setting the second angle equal to the first gives the double and triple angle rules — the doubled-angle results, needed for Example 22 and several exercises
• Solving a pair of simultaneous equations for the two original unknowns
• Spotting and cancelling a common factor in a quotient

What they should be able to do

• Produce the four raw equations by adding and subtracting the expansions in pairs
• Carry out the change of variable to the half-sum and half-difference, and say why it is needed
• State all four parts of result 20, including the sign that sits outside one of them
• State result 21 and explain that it is result 20 read in the other direction
• Recognise from the shape of an expression which of the four applies
• Convert a quotient of two such sums into a single ratio by cancelling a common factor
• Handle a three-term expression by pairing the outer two first
• Decide, for a given identity, which direction of conversion to use

Where it usually goes wrong

• "The sum of two sines is the sine of the sum." It is not, and the counter is immediate: two right angles each have sine 1, adding to 2, while the sine of a straight angle is 0.
• "The 2 out in front is decoration." Drop it and the identity is false by a factor of two everywhere. Check it once by setting the second angle to zero: the half-sum and the half-difference then both become half the first angle, and the statement collapses onto the doubled-angle result, where the 2 is visibly load-bearing. Do not test it at a pair of equal angles instead — the half-difference is then zero, its cosine is 1, and what remains is the empty claim that two copies of a term add to twice that term.
• "The difference of two cosines is twice a sine times a sine." There is a minus sign outside, and it is the single most frequently dropped sign in the chapter. Verify it once with two easy angles before trusting it.
• "The half-sum and half-difference always carry the same functions." They do not, and the pattern in section 8 is worth more than the four formulas. Added terms and subtracted terms behave differently.
• "Always convert sums into products." Example 19 goes the other way first. The question to ask is which direction produces a common factor, and sometimes the answer is to expand before collecting.
• "A three-term expression cannot be converted." Pair the two outer terms, and the factor that appears is usually shared with the middle one. That is the whole method for Exercise 3.3 questions 14 and 21 and Miscellaneous questions 5 and 7.
• "These are new results, unrelated to the start of the section." They are results 3, 4, 7 and 8 added and subtracted. Miscellaneous question 4 even reproduces the chord computation that proved result 3 in the first place.

Questions to check understanding

• Convert a sum or difference of two sines or cosines into a product
• Convert a product into a sum or difference
• Prove an identity in which a quotient of two such sums reduces to a single ratio
• Prove an identity involving three or four terms by pairing them appropriately
• Show that a difference of squared sines equals a product of two sines
• Evaluate a numerical combination at unfamiliar fractions of a half turn
• Decide, given an identity to prove, which direction of conversion to attempt first

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

• The four raw equations (§3.4, p. 63). The chapter numbers results 3 and 4 as its working equations (1) and (2) and adds them, then subtracts them; it does the same to results 7 and 8 as (5) and (6). Verified: adding the two cosine expansions kills the sine terms and leaves twice the product of the two cosines; subtracting kills the cosine terms and leaves minus twice the product of the two sines; adding the two sine expansions leaves twice the sine of the first times the cosine of the second; subtracting leaves twice the cosine of the first times the sine of the second. Four equations, no new geometry, and the whole topic is already present in them.
• Why they are not yet usable (an added framing). Verified: those four are stated in terms of A and B, but their left-hand sides involve the sum and the difference of A and B. A student holding a sum of two cosines does not have A and B; they have the two angles that appear on the left. The change of variable exists to fix exactly that mismatch.
• The change of variable (§3.4, p. 63). The chapter names the sum and the difference as two new symbols, solves for the originals, and substitutes. Verified: if the two new angles are P and Q, then the old first angle is their half-sum and the old second is their half-difference. Renaming the new pair back to the ordinary letters gives result 20 as printed.
• Result 20, four parts (§3.4, pp. 63–64). Verified: the sum of two cosines is twice the cosine of S times the cosine of D; the difference of two cosines is minus twice the sine of S times the sine of D; the sum of two sines is twice the sine of S times the cosine of D; the difference of two sines is twice the cosine of S times the sine of D.
• The pattern (an added reading of result 20). Verified against all four parts: when the two terms are added, the half-difference carries a cosine and the half-sum keeps the original function; when they are subtracted, the half-difference carries a sine and the half-sum switches to the other function. Two sentences in place of four formulas.
• Result 21 (§3.4, p. 64, printed under the Remark). The same four run the other way: twice a product of two cosines is a sum of two cosines; minus twice a product of two sines is a difference of two cosines; twice a sine times a cosine is a sum of two sines; twice a cosine times a sine is a difference of two sines. Verified: these are the four raw equations from p. 63 before the change of variable, which is why the chapter can call them part of result 20 rather than a new derivation.
• Example 15 (pp. 65–66). Show that the cosines at a quarter of a half turn plus and minus the same angle add to √2 times the cosine of that angle. Inputs only. Verified: the half-sum is a quarter of a half turn and the half-difference is the angle itself, so part (i) gives twice the cosine of a quarter of a half turn times the cosine of the angle, and that leading factor is √2.
• Example 16 (p. 66). Show that the sum of the cosines at seven and five times an angle, over the difference of the sines at the same two, equals the cotangent of the angle. Inputs only. Verified: both half-sums are six times the angle and both half-differences are the angle itself, so the factor at six times the angle cancels top and bottom and the cotangent is left.
• Example 17 (p. 66). Show that the sines at five and one times an angle, less twice the sine at three times it, over the difference of the cosines at five and one times it, equals the tangent of the angle. Inputs only. Verified: pairing the outer two sines gives twice the sine at three times the angle times the cosine at twice it, so the numerator factors; the denominator converts to minus twice a product of two sines; cancelling and using the doubled-angle results reduces the whole thing to the tangent.
• Example 19 (pp. 68–69). Show that a difference of two cosine products equals a product of two sines. Inputs only. Verified: the printed route goes the other way first — result 21 turns the two products into four cosine terms, exactly one pair of which cancels, and result 20 then turns the surviving two back into a product. Count them: four out, two gone, two left. This is the example that proves the two directions are one tool.
• Example 22 (p. 71). Show that the squared cosines at an angle and at that angle plus and minus a third of a half turn add to 3/2. Inputs only. Verified: the doubled-angle face turns each square into something linear, result 20 collects the two shifted cosines into a single product — sum to product, which is result 20's direction, not result 21's — and the cosine of a third of a half turn supplies the 1/2 that cancels the remaining term.
• Exercise 3.3, questions 12 to 21 (p. 67). Hand the data over intact. Q12 a difference of squared sines at six and four times an angle; Q13 a difference of squared cosines at two and six times it; Q14 a three-term sum of sines at two, four and six times it, with the middle doubled; Q15 an identity between two cotangent-times-bracket products at four times the angle and at the angle; Q16 a quotient of a cosine difference at nine and five times it over a sine difference at seventeen and three times it; Q17 a quotient of sums at five and three times it; Q18 a quotient in two general angles; Q19 a quotient of sums at one and three times it; Q20 a quotient with a difference of squares underneath; Q21 a three-term quotient at four, three and two times it. Verified, as working added here: Q12 and Q13 both come from the observation that a difference of squared sines equals the sine of the sum times the sine of the difference, which follows from parts (iii) and (iv) multiplied together. Q21 is the three-term case: pair the outer terms in each of the numerator and the denominator, and a common bracket appears and cancels, leaving the cotangent at three times the angle.
• Miscellaneous Exercise, questions 1 to 7 (pp. 71–72). Hand the data over intact: Q1 a numerical combination at thirteenths of a half turn; Q2 a two-bracket sum in an angle and three times it; Q3 and Q4 two squared-bracket sums; Q5 a four-term sum of sines at one, three, five and seven times an angle; Q6 a quotient of four-term sums; Q7 a three-term combination. Verified, as working added here: Q4 expands to 2 less twice the cosine of the difference, which is the very chord computation that proved result 3 in One distance calculation on the unit circle yields the cosine of a difference. Q5 pairs the outermost and innermost sines separately, each pairing producing a factor at four times the angle. Q1 comes out to zero because the product converts to two cosines that are the exact negatives of the two standing beside them.

Figures to have open

• A derivation board showing the four expansions stacked in two pairs, with addition and subtraction performed in place so the cancelling terms are visible. An added construction; §3.4 prints no figure after Fig 3.14.
• A change-of-variable panel: two angles in, their half-sum and half-difference out, and the reverse. An added construction, carrying section 5.
• A four-cell pattern chart for result 20, arranged so that added pairs sit in one row and subtracted pairs in the other, making the pattern of section 8 visible at a glance. An added construction.
• A worked-quotient movement for Example 16 in which the common factor physically lifts out of both lines. The chapter prints no figure for any example.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.4, result 20 with its derivation, pp. 63–64
• §3.4, result 21, p. 64, printed under the Remark
• Examples 15, 16 and 17, pp. 65–66; Example 19, pp. 68–69; Example 22, p. 71
• Exercise 3.3, questions 12 to 21, p. 67
• Miscellaneous Exercise on Chapter 3, questions 1 to 7, pp. 71–72
• The chapter Summary, p. 74, reprints both results in full