Sine and tangent go back to the definition, because no rule so far reaches them

What to assume they know

• The rate of change at a point, defined as a limit of average rates — Definition 2 and computing from first principle
• Rules for differentiating a sum, a product and a quotient — the product and quotient rules
• Trapping a function between two others to settle the trigonometric cases — the limits of sin x over x and of (1 − cos x)/x
• The identities for a difference of two sines and for a sine of a sum, from One distance calculation on the unit circle yields the cosine of a difference
• The definitions of tan, cot, sec and cosec in terms of sin and cos
• The Pythagorean identity relating sin², cos² and 1

What they should be able to do

• Explain why no rule established so far yields the derivative of sin x, and therefore why the definition must be used
• Compute the derivative of sin x from first principle, naming the identity used and the standard limit that closes the argument
• Differentiate cos x from the definition alone, by the route analogous to the one used for sine
• Compute the derivative of tan x from first principle, and again by the quotient rule, and check the two agree
• Derive the derivatives of sec x, cosec x and cot x from those of sin and cos by the quotient rule
• Identify which of the two standard trigonometric limits each computation consumes
• Explain why sin 2x and sin²x, despite Example 18's answer, are two different differentiation problems
• Recognise the point at which each derived function is undefined, and connect that to the original function's own exclusions

Where it usually goes wrong

• "The quotient rule gives you sin x, since sin = tan × cos." It gives you nothing until you already have two of the three, and the only route into the family is through the definition. The chapter enters through sine and works outward.
• "The derivative of sin x is cos x because the graphs look shifted." That is an observation, not a derivation, and it silently assumes radian measure. In degrees the derivative of sin x is not cos x, because the standard limit is not 1. The whole of §12.4 exists to pin that down.
• "Example 17 uses the quotient rule." It does not; the chapter works tan x from the definition. The quotient-rule route is available and shorter, and a question that says from first principle rules it out.
• "sin²x and sin 2x differentiate to the same thing, since Example 18's answer is sin 2x." Example 18 differentiates the square and gets sin 2x. Example 21 differentiates sin 2x and gets 2(cos²x − sin²x). Run them adjacently.
• "The half-angle in Example 16 is a trick." It is forced: the identity for a difference of sines produces a half-argument, and the standard limit needs the denominator to match the sine's argument. The rewriting is bookkeeping to make the two agree.
• "Once you have the rules you never need first principle again." The Miscellaneous Exercise opens with four first-principle items, two of them trigonometric. The definition is the examinable fallback, not a historical curiosity.
• "The derivative of sec x is sec x tan x, so it is a standard result in this chapter." It is a correct answer to Exercise 12.2 q11 (ii), which the student is expected to derive by the quotient rule. It is not among the standard derivatives listed in the chapter's own Summary on p. 255, which prints only the power rule and the derivatives of sine and cosine.

Questions to check understanding

• Find the derivative of sin x, cos x or a shifted version from first principle
• Find the derivative of tan x from first principle
• Differentiate sec x, cosec x or cot x using the quotient rule
• Differentiate a combination such as 3 cot x + 5 cosec x
• Differentiate a product of a polynomial with a trigonometric function
• Two-mark: name the standard limit used at the final step of a supplied first-principle computation
• Explain why these derivative formulas require angles in radians

Examples worth working on the board

Values marked verified are worked out here from the chapter's printed data; no answer key was consulted.

• The gap (section 1; an added framing, drawn from what the chapter has and has not proved by p. 247). Available at this point: the derivative of a constant, of x, of any whole-number power, sums, differences, products, quotients. Nothing in that list produces sin x, because sin x is not built from powers of x by finitely many of those operations. Hence Definition 2.
• Example 16 (§12.5.2, p. 247), the derivative of sin x. The chapter's route: form the difference quotient; convert the difference of the two sines into a product using the identity for that difference; the result carries a cosine whose argument is the original point shifted by half the increment, and a sine of half the increment divided by the increment. Rewriting the second factor so that its denominator is half the increment makes it one of §12.4's standard limits, which tends to 1. The cosine factor tends to cos x. Verified: the product of the two limits is cos x times 1, so the derivative of sin x is cos x. Note that the halving is forced by the identity, not chosen.
• Exercise 12.2 q10 (p. 249) asks for cos x to be differentiated from the definition alone. Verified by the same route: the corresponding identity for a difference of two cosines produces a sine of the shifted point times the same half-angle factor, with an overall minus sign, and the standard limit closes it the same way. The answer is −sin x. The chapter does not work this one; the Summary on p. 255 states the result.
• Example 17 (pp. 247–248), the derivative of tan x from first principle. The chapter's route: form the difference quotient, combine the two tangents over a common denominator of two cosines, recognise the resulting numerator as a sine of the difference of the two angles — which is just the increment — and split the limit into the standard sine limit times the reciprocal of a product of two cosines. Verified: the first factor tends to 1 and the second to the reciprocal of the square of cos x, so the derivative is sec²x.
• The quotient-rule route to tan x (section 7; not the chapter's own computation, but licensed by the material on p. 244 together with Examples 16 and q10). Verified: applying the quotient rule to sin over cos gives a numerator of cos²x + sin²x, which the Pythagorean identity turns into 1, over cos²x — the same sec²x. Present this after Example 17, not instead of it; the chapter's own route is the examinable one for the "from first principle" phrasing.
• Example 21 (ii) (pp. 251–252), the derivative of cot x, computed twice — once as cos over sin, once as 1 over tan. Both give −cosec²x. This is worked in detail in Rules for differentiating a sum, a product and a quotient; recall it here as the model for the reciprocal family.
• Exercise 12.2 q11 (p. 249) sets seven trigonometric derivatives. Item (i) is sin x cos x, which the product rule handles. Item (ii) is sec x, and item (iv) is cosec x — each is the quotient rule applied to 1 over a named function. Item (iii) is 5 sec x + 4 cos x, and item (v) is 3 cot x + 5 cosec x; both are assembled by the sum rule with constants pulled out. Item (vi) is 5 sin x − 6 cos x + 7, where the trailing constant contributes nothing. Item (vii) is 2 tan x − 7 sec x, which needs Example 17's result as one of its two pieces.
• Example 18 (p. 248), the derivative of sin²x by the product rule, giving twice sin x cos x, which the chapter rewrites as sin 2x.
• Example 21 (i) (p. 251), the derivative of sin 2x, obtained by rewriting it as twice sin x cos x first and then applying the product rule, giving 2(cos²x − sin²x). Verified: the two examples use the double-angle identity in opposite directions and produce different answers, which is exactly the confusion worth heading off.
• Example 20 (pp. 250–251), two first-principle computations the chapter sets out in full: sin x + cos x, giving cos x − sin x; and x sin x, giving x cos x + sin x. Both consume both of §12.4's standard limits — the sine limit and the cosine-difference limit — which makes them a good closing check that the student can spot which limit is being used where.
• Miscellaneous Exercise on Chapter 12, item 1 (p. 253) parts (iii) and (iv): sin(x + 1) and cos(x − π/8), both from first principle. These need the same sum-to-product move as Example 16, with the shift carried along.
• Where the derivatives fail to be defined (section 10). tan x and sec x are undefined where cos x is 0, and so are their derivatives; cot x and cosec x are undefined where sin x is 0, and so are theirs. sin x and cos x are defined everywhere and so are their derivatives. Verified by inspection of each formula.

Figures to have open

• A containment diagram: constants, powers, and everything built from them by the four rules, drawn as a region with sin and cos plotted outside it and one arrow labelled definition crossing the boundary. Standard schematic; §12.5.2 prints no figure, which I confirmed on pp. 246, 247 and 248.
• A step-by-step reveal of the sine computation with the §12.4 limit shown arriving from outside the frame, so its provenance is visible.
• A two-column agreement panel for tan x by two routes.
• A number line marked with the multiples of π/2, colour-coded for where the tangent family and the cotangent family each fail.
• No textbook figure is needed for this topic.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 12 "Limits and Derivatives", §12.5.2 Derivative of polynomials and trigonometric functions, Example 16 (p. 247), Example 17 (pp. 247–248), Example 18 (p. 248).
• Miscellaneous Examples: Example 20 (pp. 250–251), Example 21 (p. 251), Example 22 (pp. 252–253).
• Exercise 12.2 items 10 and 11, printed p. 249; Miscellaneous Exercise on Chapter 12 item 1 parts (iii) and (iv), p. 253, and the trigonometric items 14–30, pp. 253–254.
• Chapter Summary, p. 255, which lists the derivatives of sine and cosine.
• Backward references inside the chapter: the two standard limits are Theorem 5, §12.4, p. 235; the rules used are on p. 244.
• Deliberate cross-reference outside this chapter: the sum, difference and double-angle identities are Chapter 3's, used here without re-derivation.