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Circular Measurement of Angle

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Question

Show that: ${cos}^{2} α + {cos}^{2} (α + β) - 2 cos α cos β cos (α + β) = {sin}^{2} β$ .

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Solution

${cos}^{2} α + {cos}^{2} (α + β) - 2 cos α cos β (cos (α + β))$
$= {cos}^{2} α + cos (α + β) [cos (α + β) - 2 cos α cos β]$
$= {cos}^{2} α + cos (α + β) [- cos α cos β - sin α sin β]$
$= {cos}^{2} α - (cos α cos β) (cos α cos β + sin α sin β)$
$= {cos}^{2} α - {cos}^{2} α {cos}^{2} β + {sin}^{2} α {sin}^{2} β$
$= {cos}^{2} α (1 - {cos}^{2} β) + {sin}^{2} α {sin}^{2} β$
$= {cos}^{2} α {sin}^{2} β + {sin}^{2} α {sin}^{2} β = {sin}^{2} β ({sin}^{2} α + {cos}^{2} β)$
$= {sin}^{2} β$

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