Rewriting a product of trigonometric ratios into terms you can already integrate

What to assume they know

• The standard formulae for the six trigonometric ratios, from module m01
• Substitution, from the previous topic, including the four integrals of the tangent, cotangent, secant and cosecant
• The double-angle and triple-angle identities for the sine and the cosine, from Class XI
• The four product identities that turn a product of two ratios into a sum, from Class XI
• The Pythagorean identities in all three forms, from Class XI
• That the sine is an odd function and the cosine an even one

What they should be able to do

• Say why a product of two ratios with different arguments defeats substitution, and what has to happen first
• Use a double-angle identity to reduce a squared ratio to a first power
• Use a product identity to turn a product of two ratios into a sum of two, and account for the sign the oddness of the sine introduces
• Use the triple-angle identity to reduce a cubed sine, and integrate the result
• Integrate the same cubed sine by substitution instead, and reconcile the two answers exactly
• State what the chapter's closing Remark claims and carry out the check it leaves to the reader
• Recognise which identity family an exercise item is asking for, from the shape of the integrand
• Identify an item whose statement needs a restriction the exercise does not print

Where it usually goes wrong

• "Every trigonometric integral needs an identity." Exercise 7.3 Q17 and Q18 need only a split, and the previous topic's Example 6 needed only a substitution. The identity is for when the integrand's arguments do not match or its powers are too high.
• "The identity is the answer." It is the first line. Every item in this subsection ends with a lookup in the module m01 list, and a script that celebrates the identity and hurries the integration has inverted the emphasis.
• "The identity gives a sum of two sines, so the working should show a plus." It shows a subtraction because the second argument is negative and the sine is odd. This is a one-line step the chapter performs without saying so, and it is the likeliest place in the subsection for a student to conclude the book has made a sign error.
• "Two different answers to the same integral means one is wrong." Module m01 settled this: two anti derivatives differ by a constant. Here they do not even do that — the two answers to Example 7 (iii) are equal outright, and showing it is a good use of ninety seconds.
• "The bracketed prompt is rhetorical." It is a question with an answer, and the answer is the sum and difference formulas for the sine added together. Leaving it unanswered trains students to skip the chapter's own invitations.
• "Reducing the power is always better than substituting." Example 7 (iii) does both and the substitution is shorter. The identity route generalises better to even powers, where no factor can be peeled off; say which route suits which parity rather than declaring a winner.
• "An inverse ratio inside an integrand behaves like any other function." Exercise 7.3 Q21 puts an inverse sine around a cosine, and the simplification every solver reaches for is only valid on a restricted range that the exercise does not state. See section 10 and the note below.
• "Three factors just means applying the product identity three times." It means applying it twice and then integrating three terms; the count of applications is one less than the count of factors. Q3 and Q6 are where this bites.

Questions to check understanding

• Integrate a squared trigonometric ratio by reducing its power
• Integrate a product of two ratios with different arguments — the form of Example 7 (ii) and of Exercise 7.3 Q2 and Q7
• Integrate a product of three ratios, applying the product identity twice — the form of Exercise 7.3 Q3 and Q6
• Integrate an odd power of a sine or cosine by both available routes, and show the two answers agree
• Split a sum of cubes over a product of squares into standard rows — the form of Exercise 7.3 Q17
• Justify the product identity from the sum and difference formulas
• Choose the correct integral from four options — the form of Exercise 7.3 Q23 and Q24
• State the restriction under which an inverse sine of a cosine simplifies

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• §7.3.2's framing (Part II p. 241). Two lines: when the integrand carries trigonometric functions, known identities are used to reach the integral, and one example follows. That is the entire prose of the subsection. Everything else on Part II pp. 241–242 is worked mathematics, and everything on Part II p. 243 is the exercise. The explanation therefore has to supply the organising idea itself; section 2 and section 11 are where it does.
• Example 7 (i) (Part II pp. 241–242). The square of the cosine. The chapter recalls the double-angle form that expresses the cosine of twice an angle through the square of the cosine, rearranges it to isolate the square, and integrates the two resulting terms separately. Verified: the answer is half the variable plus a quarter of the sine of twice it. Read off the page image, including the fraction denominators.
• Example 7 (ii) (Part II p. 242). A sine of twice the variable times a cosine of three times it. The chapter recalls the identity expressing such a product as half a sum of two sines, one of the sum of the arguments and one of their difference, and appends a bracketed prompt asking the reader why. Applying it here gives a sine of five times the variable and a sine of minus the variable; the second is turned into a minus by the oddness of the sine, and the chapter's next line already carries that minus. The oddness step is silent and should not be — it is the whole reason the printed working shows a subtraction where the identity shows an addition. Verified: the answer is minus a tenth of the cosine of five times the variable plus half the cosine of the variable.
• Example 7 (iii), both routes (Part II p. 242). The cube of the sine. Route one, by identity: the triple-angle form is rearranged to isolate the cube, and the two terms are integrated separately. Verified: the answer is minus three quarters of the cosine plus one twelfth of the cosine of three times the variable. Route two, marked Alternatively and done by substitution: peel one sine off, convert the remaining square by the Pythagorean identity, substitute for the cosine. Verified: the answer is minus the cosine plus a third of its cube. Both were read off the page image.
• The Remark (Part II p. 242). One sentence saying the two answers can be shown equivalent using trigonometric identities, and leaving the showing to the reader. Do the showing. Verified: expanding the cosine of three times the variable through the cube of the cosine turns one twelfth of it into a third of the cube less a quarter of the cosine; adding that to minus three quarters of the cosine gives exactly minus the cosine plus a third of the cube. The two agree exactly, with no constant left over — which is a stronger and more surprising statement than the Remark's word equivalent promises, since module m01 has taught the class to expect a constant discrepancy. That contrast is section 7.
• Exercise 7.3 (Part II p. 243), twenty-four items, of which the last two are multiple choice. Grouped by what each one wants — an added classification, not the chapter's: Q1, Q10, Q11 and Q12 are power reduction on a square or a fourth power; Q2, Q3, Q6 and Q7 are the product identity, with Q3 and Q6 needing it applied twice because they carry three factors where Example 7 (ii) carries two; Q4, Q5, Q9, Q15 and Q16 peel a factor off and use a Pythagorean identity, the Example 7 (iii) route; Q8, Q14, Q17, Q18 and Q19 are algebraic rewrites into standard rows with no identity needed beyond the Pythagorean; Q13 factorises a difference of two double-angle expressions; and Q20 and Q22 both need a factor manufactured, in the manner of Example 6 (iii) of the previous topic.
• Exercise 7.3, question 17 (Part II p. 243), worth calling out because it is the cleanest illustration of section 11. The integrand is a sum of two cubes over a product of two squares. Verified: splitting it term by term gives the secant times the tangent plus the cosecant times the cotangent, both of which are rows of the module m01 list, so the answer is the secant less the cosecant. No identity is used at all — only a split — and the item still belongs here because seeing the split requires the same instinct.
• Exercise 7.3, questions 23 and 24 (Part II p. 243). Two multiple-choice items. Verified by working added here: Q23's integrand is a difference of two squares over their product, which separates into the square of the secant less the square of the cosecant, giving the tangent plus the cotangent, which is the first option. Q24's integrand is the derivative of a product divided by the square of the cosine of that product, so the answer is the tangent of the product, which is the second option. The chapter prints no answers.
• The Summary (Part II p. 289). Identities are mentioned in one clause inside the substitution bullet and this subsection has no bullet of its own. Worth knowing when planning revision: a student working from the Summary meets §7.3.2 only as a subordinate clause.

Figures to have open

• A grouping table of the twenty-four Exercise 7.3 items by identity family, for section 9, and a three-row table of the identity families for section 2. The items and the identities are the chapter's own, from Part II pp. 241–243; the groupings are added here. Build both with the repo's DataTable component.
• A side-by-side reconciliation of the two answers to Example 7 (iii) for section 7, with the expansion of the cosine of three times the variable shown as a middle column. Both answers are the chapter's own, on Part II p. 242; the reconciliation is added here, and the chapter's Remark explicitly leaves it to the reader.
• A number line marking the interval on which an inverse sine of a cosine simplifies to a linear expression, for section 10. Not in the book; neither this chapter nor Exercise 7.3 draws or states it.
• No figure is available from the chapter for any section of this topic. The whole chapter prints one figure, on Part II p. 267, and it belongs to module m03. Every picture listed here is an added construction over the chapter's own content.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 7 "Integrals", §7.3.2 Integration using trigonometric identities, Part II p. 241
• Example 7, three parts, with the bracketed prompt and the alternative route, Part II pp. 241–242
• The Remark on the equivalence of the two answers, Part II p. 242
• Exercise 7.3, questions 1 to 24, Part II p. 243
• Summary, the clause on identities inside the substitution bullet, Part II p. 289