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The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
Explain why the length of \(C D\) can be written as \(a \cos y\).
Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).