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Show that \[ \sin(x+y) -\sin(x-y) = 2 \cos x \, \sin y \] and deduce that \[ \sin A - \sin B = 2 \cos \tfrac12 (A+B) \, \sin\tfrac12 (A-B) \,. \] Show also that \[ \cos A - \cos B = -2 \sin \tfrac12(A+B) \, \sin\tfrac12(A-B)\,. \] The points \(P\), \(Q\), \(R\) and \(S\) have coordinates \(\left(a\cos p,b\sin p\right)\), \(\left(a\cos q,b\sin q\right)\), \(\left(a\cos r,b\sin r\right)\) and \(\left(a\cos s,b\sin s\right)\) respectively, where \(0\le p < q < r < s <2\pi\), and \(a\) and \(b\) are positive. Given that neither of the lines \(PQ\) and \(SR\) is vertical, show that these lines are parallel if and only if \[ r+s-p-q = 2\pi\,. \]
Solution: \begin{align*} && \sin(x+y) - \sin(x-y) &= \sin x \cos y + \cos x \sin y - (\sin x \cos y - \cos x \sin y )\\ &&&= 2 \cos x \sin y \\ \\ && A &= x+y \\ && B &= x - y \\ \Rightarrow && x = \frac12(A+B) &\quad y = \frac12(A-B) \\ \Rightarrow && \sin A - \sin B &= 2 \cos \tfrac12(A+B) \sin \tfrac12(A-B) \\ \\ && \cos (x+y) - \cos (x-y) &= \cos x \cos y - \sin x \sin y -(\cos x \cos y + \sin x \sin y ) \\ &&&= -2 \sin x \sin y \\ \Rightarrow && \cos A - \cos B &= - 2 \sin \tfrac12 (A+B) \sin \tfrac12 (A-B) \end{align*} \begin{align*} && \text{Gradient of }PQ &= \frac{b \sin q - b \sin p}{a \cos q - a \cos p } \\ && \text{Gradient of }SR &= \frac{b \sin s - b \sin r}{a \cos s - a \cos r} \\ PQ \parallel SR \Rightarrow && \frac{b \sin q - b \sin p}{a \cos q - a \cos p } &= \frac{b \sin s - b \sin r}{a \cos s - a \cos r} \\ \Rightarrow && (\sin q - \sin p)(\cos s - \cos r) &= (\sin s - \sin r)(\cos q - \cos r) \\ \Rightarrow && -4 \cos \tfrac12(p+q) \sin\tfrac12(q-p) \sin \tfrac12(s+r) \sin \tfrac12(s-r) &= -4 \cos \tfrac12(s+r) \sin \tfrac12(s-r) \sin \tfrac12 (p+q) \sin\tfrac12 (q-p) \\ \Rightarrow && 0 &= \sin \tfrac12(s-r)\sin\tfrac12(p-q) \left ( \cos \tfrac12(p+q)\sin \tfrac12(s+r) - \sin \tfrac12 (p+q)\cos \tfrac12(s+r) \right) \\ &&&= \sin \tfrac12(s-r)\sin\tfrac12(p-q) \sin \left ( \frac12 (s+r -(p+q))\right) \end{align*} Since \(s \neq r\) and \(p \neq q\) (neither line vertical) we must have \(\frac12 (s+r -(p+q)) = n \pi \Rightarrow s+r - p - q = 0, 2\pi, 4\pi, \cdots\) but given the range constraints it must be \(2 \pi\)