Recall that #tanx# can be written as #sinx/cosx#:
#(sin2x)/(cos2x) = 2cosx#
#sin2x = 2cosx(cos2x)#
#(sin2x)/(2cosx) = cos2x#
#sin2x# can be written as #2sinxcosx#
#(2sinxcosx)/(2cosx) = cos2x#
#sinx= cos2x#
#cos2x# can be expanded as #cos^2x - sin^2x#.
#sinx = cos^2x- sin^2x#
#sinx = 1 - sin^2x - sin^2x#
#sinx = 1 - 2sin^2x#
#2sin^2x + sinx - 1 = 0#
Let #t = sinx#:
#2t^2 + t - 1 = 0#
#2t^2 + 2t - t - 1 = 0#
#2t(t + 1) - (t + 1) =0#
#(2t - 1)(t + 1) = 0#
#t = 1/2 and -1#
#sinx = 1/2 and sinx= -1#
#x = pi/6, (5pi)/6 (3pi)/2#
Hopefully this helps!
