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.
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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
- 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
14 - 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
- Explain why the argument used in part (a) only proves the identity when \(A\) and \(B\) are acute angles.
14
- 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 \(\sin ( A - B )\).