How do you verify #2(tan(2A)) * (2(cos^2(2A) - sin^2(4A)) = sin(8A)#?

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Answer 1 · 2016-04-15T20:54:57

shown below

#2tan(2A)xx2[cos^2 (2A)-sin^2 (4A)]=sin(8A)#

LHS=left hand side and RHS=right hand side. So I start with the left hand side and show that it equals the right hand side.

#LHS=2tan(2A)xx[2cos^2 (2A)-2sin^2 (4A)]#

#=4tan(2A)cos^2 (2A)-4tan2Asin^2 (4A)#

#=4(sin(2A))/cos(2A) cos^2(2A)-4(sin(2A))/cos(2A) sin^2 (4A)#

#=4sin(2A)cos(2A)-4(sin(2A))/cos(2A) sin^2(2(2A))#

#=2*2sin(2A)cos(2A)-4(sin(2A))/cos(2A) xx2sin^2(2A)cos^2(2A)#

#=2sin(2(2A))-4(sin(2A)) xx2sin^2(2A)cos(2A)#

#=2sin(4A)-4*2sin(2A)cos(2A) xxsin^2(2A)#

#=2sin(4A)-4sin(4A)sin^2(2A)#

#=2sin(4A)[1-2sin^2(2A)]#

#=2sin(4A)cos2(2A)#

#=2sin(4A)cos(4A)#

#=sin(2(4A))#

#=sin(8A)#

#=RHS#