1+sin 3(a+π/2)*cos2a+2sin3a*cos(3π-a)*sin(a-π)=2sin^2(5a/2 Доказать тождество

Question

1+sin 3(a+π/2)*cos2a+2sin3a*cos(3π-a)*sin(a-π)=2sin^2(5a/2

Доказать тождество

Answer 1

Доказать тождество :
1+ sin3(α +π/2)*cos2α +2sin3α*cos(3π -α)*sin(α -π) =2sin²5α/2.
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1+ sin3(α +π/2)*cos2α +2sin3α*cos(3π -α)*sin(α -π) =
1- cos3α*cos2α + 2sin3α*(-cosα)*(-sinα) = 1- cos3α*cos2α + sin3α*sin2α =
 1- (cos3α*cos2α - sin3α*sin2α) = 1 -cos(3α+2α) =1 - cos5α = 2sin²5α/2.