NSW Y11 Maths - Extension 1 Inverse Trig Functions and Identities Trig Products as Sums or Differences

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NSW Syllabus Reference: ME-T2 Further Trigonometric Identities. This will require student to

derive and use the sum and difference expansions for the trigonometric functions \(\sin (𝐴 \pm 𝐵),\, \cos (𝐴 \pm 𝐵)\) and \(\tan (𝐴 \pm 𝐵)\) (ACMSM044)
derive and use the double angle formulae for \(\sin 2𝐴,\, \cos 2𝐴\) and \(\tan 2𝐴\) (ACMSM044)
derive and use expressions for \(\sin 𝐴,\, \cos 𝐴\) and \(\tan 𝐴\) in terms of \(t\) where \(t=\tan \dfrac{A}{2}\) (the \(t\)-formulae)
derive and use the formulae for trigonometric products as sums and differences for \(\cos 𝐴 cos 𝐵,\, sin 𝐴 sin 𝐵,\, sin 𝐴 cos 𝐵\) and \(cos 𝐴 sin 𝐵\) (ACMSM047)

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Trig Products as Sums or Differences - Video - Product To Sum Identities and Sum To Product Formulas

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Formulae for products as sums or differences:

\(\sin A\cos B = \dfrac{1}{2}\left( {\sin \left( {A + B} \right) + \sin \left( {A - B} \right)} \right)\)
\(\cos A\sin B = \dfrac{1}{2}\left( {\sin \left( {A + B} \right) - \sin \left( {A - B} \right)} \right)\)
\(\cos A\cos B = \dfrac{1}{2}\left( {\cos \left( {A + B} \right) + \cos \left( {A - B} \right)} \right)\)
\(\sin A\sin B = \dfrac{1}{2}\left( {\cos \left( {A - B} \right) - \cos \left( {A + B} \right)} \right)\)

Formulae for sums or differences as products:

\(\sin X + \sin Y = 2\sin \left( {\dfrac{{X + Y}}{2}} \right)\cos \left( {\dfrac{{X - Y}}{2}} \right)\)
\(\sin X - \sin Y = 2\cos \left( {\dfrac{{X + Y}}{2}} \right)\sin \left( {\dfrac{{X - Y}}{2}} \right)\)
\(\cos X + \cos Y = 2\cos \left( {\dfrac{{X + Y}}{2}} \right)\cos \left( {\dfrac{{X - Y}}{2}} \right)\)
\(\cos X - \cos Y = - 2\sin \left( {\dfrac{{X + Y}}{2}} \right)\sin \left( {\dfrac{{X - Y}}{2}} \right)\)

Problems involving the above formulae will be presented in the subtopic.