Verify the identity. Show your work. cos(α - β) - cos(α + β) = 2 sin α sin β

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isabellahardes isabellahardes

28-03-2018
Mathematics

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Verify the identity. Show your work. cos(α - β) - cos(α + β) = 2 sin α sin β

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jdoe0001 jdoe0001 · Answer 1 · 2018-03-28T21:44:37+02:00

[tex]\bf \textit{Sum and Difference Identities} \\\\ cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\ cos(\alpha - \beta)= cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)\\\\ -------------------------------[/tex]

[tex]\bf {cos(\alpha - \beta)-cos(\alpha+\beta)}=2sin(\alpha)sin(\beta) \\\\\\ \stackrel{\textit{left-hand-side}}{[cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)]-[cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta)]} \\\\\\ \underline{cos(\alpha)cos(\beta)} + sin(\alpha)sin(\beta)-\underline{cos(\alpha)cos(\beta)} + sin(\alpha)sin(\beta) \\\\\\ sin(\alpha)sin(\beta)+ sin(\alpha)sin(\beta)\implies 2 sin(\alpha)sin(\beta)[/tex]