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

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
• Coordinates on the unit circle extend the ratios to every real number — the tangent and cotangent as quotients, and the exact inputs at which each fails to exist
• Simplifying a compound fraction by dividing numerator and denominator by the same non-zero quantity

What they should be able to do

• Derive result 10 as a quotient of results 7 and 3, and name the divisor used
• State the three angles result 10 excludes and explain what each exclusion protects
• Derive result 11 from result 10 by negating the second input
• Derive result 12 as a quotient of results 3 and 7, with the sines as divisor
• State the exclusions on results 12 and 13 and say why they differ from result 10's
• Identify the condition the printed result 11 omits and supply it
• Explain why the Summary's plus-or-minus form is the accurate statement
• Use results 10 and 11 to evaluate a tangent at a non-standard angle and to prove a stated identity

Where it usually goes wrong

• "The conditions are legal small print." They are the division step. Every one of them names an angle where something in the derivation would have been zero. If a student cannot point at the divisor a condition protects, they have not seen the proof.
• "The formula only fails where the left-hand side fails." No. In the worked case above the left side is a perfectly good 1 and the right side does not exist, because one of the two individual tangents is missing. Conditions on A and on B are not consequences of the condition on A ± B.
• "Result 11 has no conditions, because none are printed." They were inherited from result 10 through the substitution, and the Summary on p. 73 states them. This is a genuine inconsistency in the printed section, not a subtlety.
• "The cotangent version is just one over the tangent version." The cotangent of a sum is indeed the reciprocal of the tangent of a sum, but result 12 is expressed in the cotangents of A and B, which takes a second division to reach. That is why the chapter proves it separately, and why its exclusions are a different set.
• "Both sum results exclude the same angles." They do not. The tangent version excludes odd multiples of a quarter turn; the cotangent version excludes multiples of a half turn. The two lists have no members in common.
• "You can always cancel the cosines." Only when they are not zero. The whole section is an object lesson in checking a divisor before dividing by it.

Questions to check understanding

• Derive the tangent of a sum from the sine and cosine expansions, stating the divisor and the conditions
• State the exclusions on each of results 10 to 13 and justify each one
• Evaluate a tangent at an angle expressible as a sum or difference of two standard angles
• Prove an identity in which a quotient of sines is converted to a quotient of tangents
• Show that a product of three tangents equals an alternating sum of them
• Given an identity and a particular pair of angles, decide whether it applies

Examples worth working on the board

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

• Result 10 (§3.4, p. 60). Printed with its condition attached in front: none of A, B or A + B may be an odd multiple of a quarter turn. The chapter then notes that this makes the three cosines non-zero, writes the tangent of A + B as result 7 over result 3, and divides top and bottom by the product of the two cosines. Verified: what emerges is the sum of the two tangents over one minus their product.
• What each of the three exclusions protects. Verified, and this is the heart of the topic: excluding A and B keeps their own tangents in existence and makes the divisor non-zero; excluding A + B keeps the left-hand side in existence. The three are doing different jobs, and a student who can say which is which will never write the condition down from memory incorrectly.
• Result 11 (§3.4, p. 60). Obtained by negating B in result 10, so the sum of the tangents becomes a difference and the minus in the divisor becomes a plus. Verified, and worth stating plainly: the printed statement of result 11 carries no condition of its own — although the substitution inherits one. The correct condition is that none of A, B or A − B is an odd multiple of a quarter turn.
• Result 12 (§3.4, pp. 60–61). Printed with its own condition: none of A, B or A + B may be a whole-number multiple of a half turn, which is what makes the three sines non-zero. The cotangent of A + B is written as result 3 over result 7 and divided top and bottom by the product of the two sines. Verified: the outcome is the product of the two cotangents less one, over the sum of the two cotangents — and note the order in which the two cotangents appear in that sum on the printed page, which is not the order a student would guess.
• Result 13 (§3.4, p. 61). The difference form, obtained by negating B in result 12. Verified from p. 61: unlike result 11, this one does carry a printed condition — none of A, B or A − B a multiple of a half turn. So the two difference forms are stated inconsistently on facing halves of the same section.
• The Summary (p. 73). Verified from the printed page: the Summary states one condition for both tangent forms and one for both cotangent forms, using a plus-or-minus on the combined angle. That is the accurate statement of all four, and it repairs the gap in result 11.
• Where the missing condition bites (an added worked case). Verified: take A as a quarter turn and B as an eighth of a turn — that is, 90° and 45°. The left-hand side of result 11 is the tangent of 45°, which is 1. The right-hand side needs the tangent of 90°, which does not exist, so it is not a number at all. The identity as printed on p. 60 has a perfectly good left side and a meaningless right side. One slide, and the point about conditions is made permanently.
• Example 12 (pp. 64–65). Evaluate the tangent at thirteen twelfths of a half turn. Inputs only. Verified: a half turn can be peeled off by the tangent's own repetition, leaving a twelfth of a half turn, which splits as an eighth of a full turn minus a twelfth of one — 45° less 30°, in the degree names this brief uses elsewhere; result 11 with the tangents 1 and 1/√3 then gives 2 − √3.
• Example 13 (p. 65). Prove that the sine of a sum over the sine of a difference equals the sum of the two tangents over their difference. Inputs only. Verified: expand both sides with results 7 and 8 and divide numerator and denominator by the product of the two cosines — the identical move that produced result 10. Note that the chapter attaches no condition here either, although the working needs both cosines non-zero and the sine of the difference non-zero.
• Example 14 (p. 65). Show that the product of the tangents at three consecutive whole multiples of an angle equals the corresponding alternating difference. Inputs only. Verified: write the triple as the double plus the single, apply result 10, clear the fraction and rearrange. The conditions travel with result 10 and are again not stated.
• Exercise 3.3, questions on this topic (pp. 67–68). Hand the data over intact. Q7 is a quotient of the tangents at an eighth of a turn plus and minus the same angle, to be shown equal to a squared quotient in the tangent of that angle. Q22 is an alternating combination of the cotangents at an angle and its double and triple, to be shown equal to 1. Verified: Q7 falls out of results 10 and 11 with the tangent of an eighth of a turn taken as 1 — note that a quarter turn would not serve, since the tangent has no value there at all — and Q22 falls out of result 12 applied with the double and the single as the two angles.

Figures to have open

• A derivation strip for result 10: quotient, then division, then the tidy form, with the divisor kept visible throughout. An added construction; §3.4 prints no figure after Fig 3.14.
• An exclusion map: the input line with the odd multiples of a quarter turn punched out in one colour and the multiples of a half turn in another, so the two conditions are visibly different sets. An added construction, and the cleanest way to carry section 6.
• A side-by-side panel of results 10 to 13 exactly as printed, with the missing condition on result 11 marked as an omission rather than silently filled in. Standard schematic.
• A two-column evaluation of the worked case, one column reaching a number and one reaching a gap. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.4, result 10, p. 60 — the condition, the quotient and the division
• §3.4, result 11, p. 60 — the difference form, printed without a condition
• §3.4, result 12, pp. 60–61, and result 13, p. 61 — the cotangent forms, both conditions as printed
• Examples 12 and 13, pp. 64–65; Example 14, p. 65
• Exercise 3.3, question 7, p. 67, and question 22, p. 68
• The chapter Summary, p. 73, which states both conditions with a plus-or-minus on the combined angle