Exercise 1: Verifying R(30°)·R(45°)'s Top-Left Entry by Hand — Possible Solution
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WHAT THE TOP-LEFT ENTRY OF A MATRIX PRODUCT REPRESENTS
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For R(a) = [[cos a, -sin a],[sin a, cos a]] and R(b) defined the same
way, the top-left entry of the product R(a)*R(b) comes from
multiplying the first row of R(a) by the first column of R(b):

    (R(a)*R(b))[0][0] = cos(a)*cos(b) + (-sin(a))*sin(b)
                       = cos(a)cos(b) - sin(a)sin(b)

APPLYING THE ANGLE-ADDITION IDENTITY
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The trigonometric identity given, cos(a+b) = cos(a)cos(b) -
sin(a)sin(b), is EXACTLY the expression just derived for the
top-left entry. So algebraically:

    (R(a)*R(b))[0][0] = cos(a)cos(b) - sin(a)sin(b) = cos(a+b)

For a=30 degrees and b=45 degrees, this means the top-left entry of
R(30)*R(45) should equal cos(75 degrees).

CONFIRMING AGAINST THIS CHAPTER'S VERIFIED NUMBER
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cos(75 degrees) = 0.25881904510252074 (matching this chapter's own
verified R(75°) direct computation's top-left entry exactly). This
chapter's own verified R(30°)*R(45°) product had a top-left entry of
0.25881904510252085 - matching to about 15 significant digits, with
the tiny remaining difference (in the 16th digit) simply being
ordinary floating-point rounding from performing the matrix
multiplication as three separate floating-point operations (two
multiplications and a subtraction) rather than the trigonometric
functions being evaluated once directly.

WHY THIS CONFIRMS THE COMPOSITION RULE ISN'T A COINCIDENCE
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This derivation shows the rotation matrix's own multiplication rule
isn't just numerically observed to match angle addition - it's
algebraically IDENTICAL to the angle-addition identity, entry by
entry. The matrix multiplication mechanically reproduces the
trigonometric identity because the rotation matrix's entries were
built directly from cosine and sine in the first place.

WHY THIS WORKS AS AN ANSWER
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The answer derives the top-left entry of the matrix product
symbolically, recognizes it as exactly the angle-addition identity
rather than merely similar to it, and cross-checks the algebraic
result against this chapter's own two verified numeric values,
explaining the tiny remaining discrepancy as ordinary floating-point
rounding rather than an error.