Two more identities from the same equation, and the angles they hold for

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What to assume they know

Dividing Pythagoras through by the hypotenuse squared — the first identity, and the divide-Pythagoras-through move that produces it
The extreme cases at 0° and 90°, and the ratios that stop being defined — which ratios are undefined at 0° and at 90°, and why
Defining sine, cosine and tangent, then their three reciprocals — the reciprocal pairings, which decide which two ratios each division produces
Manipulating an algebraic fraction: converting to a common denominator, cancelling a common factor, multiplying by a conjugate
That a proof of an identity works one side down to the other rather than assuming the result

What they should be able to do

Derive the identity relating the tangent and the secant by dividing Pythagoras by the square of the adjacent leg
Derive the identity relating the cotangent and the cosecant by dividing by the square of the opposite leg
State the range of angles for each of the three identities and justify each restriction by naming the side that vanishes
Recognise that one identity may be written either as a sum or as a difference, and that these are the same statement
Use an identity to obtain the remaining ratios from one given ratio
Prove a stated identity by transforming one side into the other, choosing which ratios to convert to before starting
Choose between the three identities according to which ratios an expression already contains

Where it usually goes wrong

"There are three identities to memorise." There is one equation and three choices of divisor. A student who can divide Pythagoras by any of the three sides never has to remember which identity pairs which ratios — the division tells them.
"The range conditions are exam pedantry." They are the reason the identity is true. Each excluded endpoint is a division by a side of length zero, which is the same failure that put four gaps in Table 8.1.
"The tangent identity fails at 90°, so it gives a wrong answer there." It gives no answer there. Both ratios in it are undefined at 90°, so the statement has nothing to be true or false about.
"A sum form and a difference form are two different results." One rearrangement apart. The book prints both, in the body and in the summary, and a student meeting them a week apart will otherwise learn two things.
"Every identity proof needs an identity." Example 11 needs only factoring. Reaching for an identity before looking at the expression is the commonest way these proofs go long.
"You may work on both sides at once." Two of the exercise hints explicitly permit simplifying each side separately, which is a different discipline from assuming what you are proving and manipulating across the equals sign.
"Which identity to use is a matter of luck." It is a matter of vocabulary. Look at which ratios the target contains and convert everything into those first — that is exactly what Example 12 announces before it starts.

Questions to check understanding

Derive one of the two identities from Pythagoras and state its range
State which angles an identity excludes, and why
Express all the remaining ratios in terms of one named ratio
Multiple-choice items where an expression simplifies to a constant or to a single squared ratio
Prove a stated identity by transforming one side into the other
Prove a stated identity by simplifying each side separately to a common expression
Given an expression, say which of the three identities is the useful one and what to convert to first

Examples worth working on the board

Inputs only. Values marked verified are an added algebra on the chapter's printed data.

The second division (§8.4, p. 128). The same Pythagoras relation for Fig. 8.21, divided term by term by the square of AB — the leg adjacent to angle A. Verified: AB/AB squared is 1, BC/AB is the tangent of A, and AC/AB is the secant of A, so the result is that 1 plus the squared tangent equals the squared secant. The chapter numbers it (3).
Its range (§8.4, p. 128). The chapter checks 0° explicitly and finds it fine, then rules out 90° on the ground that neither the tangent nor the secant exists there, and states the range as 0° up to but excluding 90°. Verified against Table 8.1, p. 125: at 0° the identity reads 1 + 0 = 1; at 30° it reads 1 + 1/3 = 4/3, which is (2/√3)²; at 45° it reads 1 + 1 = 2, which is (√2)²; at 60° it reads 1 + 3 = 4, which is 2². At 90° both entries are marked undefined, so there is nothing to check — the identity has not failed, it has nothing to say.
The third division (§8.4, pp. 128–129). The same relation divided by the square of BC — the leg opposite angle A. Verified: AB/BC is the cotangent of A, BC/BC squared is 1, and AC/BC is the cosecant of A, so the squared cotangent plus 1 equals the squared cosecant. The chapter numbers it (4).
Its range (§8.4, p. 129). The cotangent and the cosecant do not exist at 0°, so the range runs from above 0° up to and including 90°. Verified against Table 8.1: at 30° it reads 3 + 1 = 4, which is 2²; at 45°, 1 + 1 = 2, which is (√2)²; at 60°, 1/3 + 1 = 4/3, which is (2/√3)²; at 90°, 0 + 1 = 1, which is 1². The pattern: identity (3) loses the top endpoint and identity (4) loses the bottom one, mirror images of each other, and each loss is the angle at which the divisor leg disappeared.
The three ranges together, as a data row. The hypotenuse identity holds from 0° through 90° with both ends included. The tangent-and-secant identity holds from 0° included up to 90° excluded. The cotangent-and-cosecant identity holds from 0° excluded up to 90° included. Verified: each excluded endpoint is precisely where the divisor side collapses, and the hypotenuse never collapses, which is why the first identity keeps both ends.
Sum form and difference form (§8.4, p. 128, against §8.5, p. 132). The body of the chapter writes the second identity as a sum equalling the squared secant; the summary writes the same thing as the squared secant minus the squared tangent equalling 1, carrying the same range. Verified: the two are one rearrangement apart.
Getting all six from one (§8.4, p. 129). The chapter's own worked chain starts from a tangent of 1/√3. Verified: the cotangent is √3; the squared secant is 1 + 1/3 = 4/3, so the secant is 2/√3 and the cosine is √3/2; the sine is the positive square root of 1 − 3/4, which is 1/2; the cosecant is 2. Cross-check against Table 8.1, p. 125 — this is the 30° column, complete.
Example 10 (§8.4, pp. 129–130). Show that a product of a secant, a bracket holding 1 minus a sine, and a bracket holding a secant plus a tangent, collapses to 1. Verified: converting all three to sines and cosines makes the two sine brackets a difference of squares, and the first identity turns that into a squared cosine, which cancels the squared cosine beneath.
Example 11 (§8.4, p. 130). A quotient whose numerator is a cotangent minus a cosine and whose denominator is the same two added, to be shown equal to the corresponding quotient in the cosecant. Verified: write the cotangent as a cosine over a sine, take the cosine out as a common factor top and bottom, and it cancels, leaving the reciprocal of the sine — the cosecant — minus and plus 1. No identity is needed at all here; the whole thing is factoring. Worth saying so, because students reach for an identity reflexively.
Example 12 (§8.4, pp. 130–131). A quotient built from a sine, a cosine and 1, to be shown equal to the reciprocal of a secant-minus-tangent, and the chapter states in advance that it will use the tangent-and-secant identity. Verified as a strategy: dividing numerator and denominator by the cosine converts the whole quotient into tangents and secants, at which point multiplying above and below by a conjugate produces a squared tangent minus a squared secant, and the identity turns that into −1. The lesson is the first move, not the algebra: look at the identity you intend to use and convert into its vocabulary before simplifying.
Exercise 8.3 (pp. 131–132). Hand the data over intact. Question 1 asks for the sine, the secant and the tangent in terms of the cotangent; verified: the sine is the reciprocal of the square root of 1 plus the squared cotangent, the tangent is the reciprocal of the cotangent, and the secant is that same square root divided by the cotangent. Question 2 asks for all the others in terms of the secant; verified: the cosine is the reciprocal of the secant, the tangent is the square root of the squared secant minus 1, the sine is that root divided by the secant, and the cotangent and cosecant are the reciprocals of the tangent and sine. Question 3 is four multiple-choice items; verified: (i) is 9, since the bracketed difference is 1; (ii) is 2; (iii) is the cosine; (iv) is the squared tangent. Question 4 sets ten identities to prove, and four of them carry a steer — but only three of those four are in brackets. Items (iii), (iv) and (ix) each end with a square-bracketed note: (iii) points at sines and cosines, and (iv) and (ix) say to work the two sides separately rather than transforming one into the other. Item (v) also names an identity to use, but it does so inside its own statement, on the same line as the thing to be proved, with no bracket around it. Item (vi) is a square root of a quotient, which the extraction loses; the radical is printed.

Figures to have open

A three-way division panel: the Pythagoras equation at the top branching into three columns, one per divisor, each resolving into its identity. Standard schematic, and it is the entire topic in one still.
A range diagram: a 0-to-90 axis with three bars, drawn with filled and open ends, each bar annotated with the side that vanishes at its open end. Standard schematic; this is the image that makes section 6 stick.
Two collapsed-triangle insets, one at each endpoint, reusing the last panels of Fig. 8.17 and Fig. 8.18 from §8.3 so the student sees the same collapse doing two jobs.
No new textbook artwork is required. Printed pp. 129, 130, 131 and 132 carry no figures at all — so sections 9 to 12 are algebra and need panels of working rather than diagrams.

Where this sits in the book

NCERT Mathematics, Textbook for Class X, Chapter 8 "Introduction to Trigonometry", §8.4, p. 128 — the division by the square of the adjacent leg, the identity numbered (3), and its range
§8.4, pp. 128–129 — the division by the square of the opposite leg, the identity numbered (4), and its range
§8.4, p. 129 — the worked chain from a tangent of 1/√3, and Examples 9 and 10
§8.4, p. 130 — Example 11, and Example 12 with its stated strategy
§8.4, p. 131 — the completion of Example 12
Exercise 8.3, pp. 131–132 — questions 1, 2, 3 and the ten items of question 4 with their three bracketed hints and the inline steer on item (v)
The chapter summary, §8.5, p. 132, point 6, which restates all three identities, giving the second in its difference form
Backward links inside this chapter: Table 8.1, p. 125, supplies every numerical check above; the undefined entries justifying both range restrictions are set out in §8.3, p. 124