Question

Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2 (α + β) = cos 2α.

Answer 1

LHS = 2 sin2 β + 4 cos (α + β) sin α sin β + cos 2(α + β)

= 2 sin2 β + 4 (cos α cos β – sin α sin β) sin α sin β + (cos 2α cos 2β – sin 2α sin 2β)

= 2 sin2 β + 4 sin α cos α sin β cos β – 4 sin2 α sin2 β + cos 2α cos 2β – sin 2α sin 2β

= 2 sin2 β + sin 2α sin 2β – 4 sin2 α sin2 β + cos 2α cos 2β – sin2α sin2β

= (1 – cos 2β) – (2 sin2 α) (2 sin2 β) + cos 2α cos 2β

= (1 – cos 2β) – (1 – cos 2α) (1 – cos 2β) + cos 2α cos 2β

= cos 2α

Answer 2

LHS

= 2 sin2 β + 4 cos (α + β) sin α sin β + cos 2(α + β) = 2 sin2 β + 4 (cos α cos β – sin α sin β) sin α sin β + (cos 2α cos 2β – sin 2α sin 2β) = 2 sin2 β + 4 sin α cos α sin β cos β – 4 sin2 α sin2 β + cos 2α cos 2β – sin 2α sin 2β = 2 sin2 β + sin 2α sin 2β – 4 sin2 α sin2 β + cos 2α cos 2β – sin2α sin2β = (1 – cos 2β) – (2 sin2 α) (2 sin2 β) + cos 2α cos 2β = (1 – cos 2β) – (1 – cos 2α) (1 – cos 2β) + cos 2α cos 2β = cos 2αRead more on Sarthaks.com - https://www.sarthaks.com/114409/show-that-2-sin-2-4-cos-sin-sin-cos-2-cos-2..

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