Quick questions on Trigonometric proofs and methods of proof in QCE Specialist Mathematics Unit 3

A direct proof of "if $P$ then $Q$" assumes $P$ and derives $Q$ through a sequence of valid steps. For example, to prove that the product of two even integers is even, write $m = 2a$ and $n = 2b$ for integers $a, b$, so $mn = 4ab = 2(2ab)$, which is even because $2ab$ is an integer. Every step must follow from a definition or an earlier line.

The statement "if $P$ then $Q$" is logically equivalent to its contrapositive "if not $Q$ then not $P$". When the negation of $Q$ is easier to work with, prove the contrapositive instead. To prove "if $n^2$ is even then $n$ is even", the contrapositive "if $n$ is odd then $n^2$ is odd" is direct: $n = 2k+1$ gives $n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$, which is odd.

Assume the statement is false and derive a contradiction. The classic example is the irrationality of $\sqrt2$: assume $\sqrt2 = \frac{p}{q}$ in lowest terms, so $p^2 = 2q^2$. Then $p^2$ is even, so $p$ is even, say $p = 2r$, giving $4r^2 = 2q^2$, hence $q^2 = 2r^2$, so $q$ is also even. But then $p$ and $q$ share the factor $2$, contradicting "lowest terms".

Not every plausible statement is true, and the correct response to a false universal claim is a single counterexample. To disprove "$\sin(A + B) = \sin A + \sin B$ for all $A, B$", take $A = B = \tfrac{\pi}{2}$: the left side is $\sin\pi = 0$ while the right side is $1 + 1 = 2$. One counterexample is a complete disproof, whereas no number of confirming examples ever proves a universal statement, which is the whole reason induction and direct proof exist.