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Circular Measurement of Angle

Prove 2sin ...

Question

Prove
$2 {sin}^{2} β + 4 cos (α + β) sin α sin β + cos 2 (α + β) = cos 2 α$

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Solution

Given $2 {sin}^{2} β + 4 cos (α + β) sin α sin β + cos 2 (α + β)$

Now $2 cos (α + β) sin β = sin (α + 2 β) - sin α$

$4 cos (α + β) sin β sin α = 2 sin (α + 2 β) sin α - 2 {sin}^{2} d$

Bur $2 sin (α + 2 β) sin α = cos (2 β) - cos [2 (α + β)]$

$∴ 2 {sin}^{2} β + 4 cos (α + β) sin α sin β + cos 2 (α + β)$

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

$= 1 - {cos}^{2} 2 β + cos 2 β - 2 {sin}^{2} α$

$= 1 - 2 {sin}^{2} α$

$= cos (2 α)$

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.

{cos}^{2} α + {cos}^{2} (α + β) - 2 cos α \cdot cos β cos (α + β) = {sin}^{2} β

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