14 Some students are trying to prove an identity for \(\sin ( A + B )\). They start by drawing two right-angled triangles \(O D E\) and \(O E F\), as shown.
\includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-22_695_662_477_689} The students' incomplete proof continues,
Let angle \(D O E = A\) and angle \(E O F = B\).
In triangle OFR,
Line \(1 \quad \sin ( A + B ) = \frac { R F } { O F }\) Line 2 $$= \frac { R P + P F } { O F }$$ Line 3 $$= \frac { D E } { O F } + \frac { P F } { O F } \text { since } D E = R P$$ Line 4 $$= \frac { D E } { \cdots \cdots } \times \frac { \cdots \cdots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5
\(=\) \(\_\_\_\_\) \(+ \cos A \sin B\) 14

1. Explain why \(\frac { P F } { E F } \times \frac { E F } { O F }\) in Line 4 leads to \(\cos A \sin B\) in Line 5
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2. Complete Line 4 and Line 5 to prove the identity Line 4 $$= \frac { D E } { \ldots \ldots } \times \frac { \cdots \ldots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 = \(+ \cos A \sin B\) 14
3. Explain why the argument used in part (a) only proves the identity when \(A\) and \(B\) are acute angles. 14
4. Another student claims that by replacing \(B\) with \(- B\) in the identity for \(\sin ( A + B )\) it is possible to find an identity for \(\sin ( A - B )\). Assuming the identity for \(\sin ( A + B )\) is correct for all values of \(A\) and \(B\), prove a similar result for \(\sin ( A - B )\).