Mathematics

Mathematics

Problem

Let α\alpha and β\beta be any real angles. Consider the product P=sin⁡(α+β) sin⁡(α−β)P=\sin(\alpha+\beta)\,\sin(\alpha-\beta). a. (i) Expand PP into a sum of terms involving sin⁡α,sin⁡β,cos⁡α\sin \alpha,\sin \beta,\cos \alpha and cos⁡β\cos \beta. b. (ii) Simplify the expanded expression to show that sin⁡(α+β) sin⁡(α−β)=sin⁡2α−sin⁡2β\sin(\alpha+\beta)\,\sin(\alpha-\beta)=\sin^{2}\alpha-\sin^{2}\beta. c. (iii) State, with reason, whether the identity obtained in part (ii) is valid for all real α\alpha and β\beta.

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