Setting the second angle equal to the first gives the double and triple angle rules

What to assume they know

• One distance calculation on the unit circle yields the cosine of a difference — results 3 and 4
• Shifts by a quarter, a half and a whole turn all fall out of the same two results — results 7 and 8
• The tangent and cotangent versions, and the angles they refuse to cover — result 10, and the habit of asking what a division assumed
• Coordinates on the unit circle extend the ratios to every real number — the first identity, which converts one squared function into the other
• Which of the six stays positive where, and what happens at the quarter turns — signs by quadrant, needed to pick roots
• Solving a quadratic by formula, for Example 20

What they should be able to do

• Derive all four faces of the doubled cosine and say which identity converts each into the next
• Derive the doubled sine, in both its forms
• Derive the doubled tangent, and read its printed condition precisely
• Derive the tripled sine and tripled cosine by splitting three into two plus one
• Derive the tripled tangent and explain why its condition covers more than the doubled tangent's does
• Choose the face of the doubled cosine that suits a given problem
• Run the doubled cosine backwards to get a half-angle value, choosing signs from the halved angle's own quadrant
• Say which conditions the chapter Summary carries and which it drops

Where it usually goes wrong

• "The doubled cosine is twice the cosine." It is not, and neither is the doubled sine twice the sine. Test both at a sixth of a half turn before anything else is said: the doubled cosine there is 1/2, while twice the cosine is √3.
• "The four faces of result 14 are four separate results." They are one result and the first identity. A student who has learnt them as four will not know which to reach for, which is the only decision this topic actually requires.
• "You can always use the tangent form." Not where the tangent does not exist, which is exactly what the conditions on results 14, 15 and 16 are saying.
• "The printed condition on result 16 makes the formula safe." It rules out the vanishing divisor and nothing else. At a quarter turn the left side is a respectable 0 and the right side is not a number. The condition on result 19 happens to cover both failures; the one on result 16 does not.
• "The triple-angle results need their own geometry." They need result 10 or result 7 with the doubled angle in one slot. Everything in §3.4 after result 9 is substitution.
• "Half-angle signs come from the original angle's quadrant." They come from the halved angle's own quadrant, which is what Example 21 establishes before it takes any square root. Halving a third-quadrant angle lands you in the second, not the third.
• "The Summary is the complete statement." The Summary on pp. 73–74 prints results 14, 15, 16 and 19 with no conditions attached at all. A student revising from it alone will not know the exclusions exist.

Questions to check understanding

• Derive all four faces of the doubled cosine and state which identity links them
• Derive the tripled sine or tripled cosine from the doubled results
• Express the tangent at four or six times an angle in terms of the tangent or cosine of the angle
• Find the sine, cosine and tangent of a halved angle from a given value and quadrant
• Find the tangent of an eighth of a half turn by solving a quadratic
• State the condition on a named double- or triple-angle result and say what it protects
• Simplify an expression containing squared sines or cosines by choosing the right face of result 14

Examples worth working on the board

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

• Result 14 (§3.4, p. 61). Setting both angles equal in result 3 gives the doubled cosine as the squared cosine less the squared sine. Verified: replacing the squared sine with one less the squared cosine gives twice the squared cosine less one; replacing the squared cosine instead gives one less twice the squared sine. That is three faces on the page, the first identity used twice. The chapter then divides the first face, written over the first identity, by the squared cosine, reaching the fourth face, entirely in the tangent — and attaches the condition that A must not be an odd multiple of a quarter turn. Verified: that is precisely where the tangent itself fails to exist, and the divisor one plus the squared tangent never vanishes, so the condition is exactly right and no more.
• Result 15 (§3.4, pp. 61–62). Setting both angles equal in result 7 gives the doubled sine as twice the product of the sine and the cosine, which needs no condition at all. Dividing by the first identity written over the squared cosine gives the tangent form, and the chapter attaches the same condition to it. Verified: again exactly the inputs where the tangent fails.
• Result 16 (§3.4, p. 62). Setting both angles equal in result 10 gives the doubled tangent as twice the tangent over one less the squared tangent, with the printed condition written on the doubled angle. Verified, and this is worth a slide: that condition rules out exactly the inputs where the divisor one less the squared tangent vanishes — and nothing else. It does not rule out A being an odd multiple of a quarter turn. Take A as a quarter turn: the left side is the tangent of a half turn, which is 0, while the right side needs the tangent of a quarter turn and so does not exist. The condition as printed is necessary but does not by itself make the right-hand side meaningful.
• Results 17 and 18 (§3.4, p. 62). The chapter writes the tripled angle as the doubled plus the single and expands with results 7 and 3 respectively, then substitutes results 14 and 15 and cleans up with the first identity. Verified: the tripled sine comes to three times the sine less four times its cube, and the tripled cosine to four times the cube of the cosine less three times the cosine. Note the pleasing near-symmetry, and note also that the signs are not mirror images.
• Result 19 (§3.4, pp. 62–63). The tripled tangent, obtained from result 10 with the doubled and the single angle, with the condition written on the tripled angle. Verified, and the contrast with result 16 is the point of the section: here the condition on the tripled angle happens to rule out both the inputs where the divisor one less three times the squared tangent vanishes and the inputs where the tangent of A itself fails. The tripled-angle condition is sufficient; the doubled-angle condition is not. That is an accident of the arithmetic, not a principle, and saying so is more useful than letting students infer a rule.
• Choosing a face (an added framing of result 14). Verified as three concrete uses: to turn a squared cosine into something linear, use the second face, the one with twice the squared cosine — that is the move Example 22 on p. 71 makes three times over, on three squared cosines; to turn a squared sine into something linear, use the third face, with twice the squared sine; and to reach a half-angle value from a known cosine, rearrange either of those two faces.
• Example 20 (p. 69). Find the tangent at an eighth of a half turn. Inputs only. Verified: doubling gives a quarter of a half turn, whose tangent is 1, so result 16 turns into a quadratic in the unknown tangent whose roots are −1 + √2 and −1 − √2; the angle lies in the first quadrant, so the value is √2 − 1. The quadrant is what discards the second root, and the chapter says so.
• Example 21 (pp. 70–71). Given a tangent of 3/4 with the angle between a half turn and three quarters of a turn, find the sine, cosine and tangent of the halved angle. Inputs only. Verified, including the three bracketed questions the page leaves open: the squared secant is 25/16 so the cosine is ±4/5, and the stated range puts the angle in the third quadrant, making it −4/5; the halved angle then lies between a quarter turn and three eighths of a turn, so its sine is positive and its cosine negative; twice the squared half-sine is one less the cosine, giving 9/5 and so a half-sine of 3/√10; twice the squared half-cosine is one plus the cosine, giving 1/5 and so a half-cosine of −1/√10; the quotient is −3. The order matters — locating the halved angle comes before any root is taken.
• Exercise 3.3, questions 23 to 25 (p. 68). Hand the data over intact. Q23 asks for the tangent at four times an angle, as a quotient in the tangent of the angle. Q24 asks for the cosine at four times an angle in terms of the sine and cosine of the angle. Q25 asks for the cosine at six times an angle as a polynomial in the cosine of the angle. Verified, as working added here: Q23 is result 16 applied to the doubled angle; Q24 follows from result 14's third face — one less twice the squared sine — applied to the doubled angle, then result 15; Q25 is result 18 applied to the doubled angle, with result 14 substituted, and the polynomial that comes out has coefficients 32, −48, 18 and −1 on the sixth, fourth, second and zeroth powers of the cosine.
• Miscellaneous Exercise, questions 8 to 10 (p. 72). Hand the data over intact: a tangent of −4/3 with the angle in the second quadrant; a cosine of −1/3 with the angle in the third; a sine of 1/4 with the angle in the second. Each asks for the sine, cosine and tangent of the halved angle. Verified, as working added here: Q8 gives 2/√5, 1/√5 and 2. Q9 gives √(2/3), −1/√3 and −√2. Q10 gives a squared half-sine of (4+√15)/8, a squared half-cosine of (4−√15)/8, both roots positive, and a halved tangent of 4 + √15. All three turn on locating the halved angle first.

Figures to have open

• A four-face panel for result 14, in the order the page prints them, with the first identity drawn as the arrow converting each of the first three into the next and the division producing the fourth. An added construction; §3.4 prints no figure after Fig 3.14.
• An input line with the exclusions of result 16 and of result 19 marked in two colours, plus the inputs where the tangent itself fails marked in a third, so the gap in result 16's condition is visible. An added construction, carrying section 5 and section 8.
• A quadrant dial that takes an angle and shows where its half lands, for section 10. An added construction, and the fastest fix for the commonest error in Example 21.
• A derivation ladder from result 3 and result 7 down to results 14 to 19, with each arrow labelled by the substitution it performs. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.4, result 14, p. 61 — the four faces, the last of them the tangent form with its condition
• §3.4, result 15, pp. 61–62, and result 16, p. 62
• §3.4, results 17 and 18, p. 62, and result 19, pp. 62–63
• Example 20, p. 69; Example 21, pp. 70–71; Example 22, p. 71, which uses the squared-cosine face three times
• Exercise 3.3, questions 23 to 25, p. 68
• Miscellaneous Exercise on Chapter 3, questions 8 to 10, p. 72
• The chapter Summary, pp. 73–74, which reprints results 14 to 19 without their conditions