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Question
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
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Solution
L.H.S = 2 sin2 β + 4 cos (α + β) sin α sin β + cos 2(α + β)
= 2 sin2β + 4(cos α cos β – sin α sin β) sin α sin β + (cos 2α cos 2β – sin 2α sin 2β)
= 2 sin2β + 4 sin α cos α sin β cos β – 4 sin2α sin2β + cos 2α cos 2β – sin 2α sin 2β
= 2 sin2 β + sin 2α sin 2β – 4 sin2α sin2β + cos 2α cos 2β – sin 2α sin 2β
= (1 – cos 2β) – (2 sin2α) (2 sin2β) + cos 2α cos 2β
= (1 – cos 2β) – (1 – cos 2α) (1 – cos 2β) + cos 2α cos 2β
= cos 2α = R.H.S.
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