Show that the following identity holds for all real and :

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Show that the following identity holds for all real α\alpha and β\beta: (i) Starting from the left-hand side, convert the product sin⁡(α+β) cos⁡(α−β)\sin(\alpha+\beta)\,\cos(\alpha-\beta) to a single trigonometric function of one angle. (ii) Similarly convert the product cos⁡(α+β) sin⁡(α−β)\cos(\alpha+\beta)\,\sin(\alpha-\beta) to a single trigonometric function of one angle. (iii) Hence prove that sin⁡(α+β) cos⁡(α−β)+cos⁡(α+β) sin⁡(α−β)=sin⁡2α.\sin(\alpha+\beta)\,\cos(\alpha-\beta)+\cos(\alpha+\beta)\,\sin(\alpha-\beta)=\sin2\alpha.

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