tan2x = 8cos^2 x - cotx
=> 2tanx/(1-tan^2 x) = 8cos^2 x - cotx
=> 2tanx = (1-tan^2 x) (8cos^2 x - cotx)
=> 2tanx = 8cos^2 x - cotx - tan^2 x * 8cos^2 x + tan^2 x * cotx
=> 2tanx = 8cos^2 x - cotx - 8 sin^2 x + tanx {since tan=sin/cos and tan = 1/cot}
=> 2tanx = 8cos2x - cotx + tanx {since, cos^2 x - sin^2x = cos2x}
=> 2tanx - tanx + cotx = 8 cos2x
=> tanx + cotx = 8 cos2x
=> sinx/cosx + cosx/sinx = 8 cos2x
=> (sin^2 x + cos^2x)/(cosx * sinx) = 8 cos2x
=> 1 = cosx*sinx * 8 cos2x
=> 1 = 4sin2x * cos2x
=> 1 = 2 sin4x {since 2sinxcosx = sin2x}
=> sin4x = 1/2
So, 4x = arcsin(1/2)
or 4x = pi/6 or 4x = 5pi/6
or x = pi/24 or x = 5pi/24