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

1 Answer
Apr 15, 2016

shown below

Explanation:

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