tan(2x)=2tan(x)/(1-tan2(x)), provided tan(x)≠±1, that is, x≠±π/4; note that when x=π/2, tan(x) is not defined but tan(2x)=0, cot(x)=0 and cos(x)=0.
2tan(x)/(1-tan2(x))=8cos2(x)-cot(x),
2tan(x)=(8cos2(x)-cot(x))(1-tan2(x)),
2tan(x)=8cos2(x)-8sin2(x)-cot(x)+tan(x)=8cos(2x)-cot(x)+tan(x),
tan(x)+cot(x)=8cos(2x),
(sin2(x)+cos2(x))/(sin(x)cos(x))=8cos(2x),
1/(sin(x)cos(x))=2/sin(2x)=8cos(2x),
2=8sin(2x)cos(2x),
4sin(4x)=2, sin(4x)=½, 4x=π/6+2nπ, 5π/6+2nπ, where n is an integer.
x=π/24+nπ/2, 5π/24+nπ/2. When n=0, x=π/24, 5π/24. When n=1, x=13π/24 which is outside the given interval.
However, returning to the original question we excluded x=±π/4, that is, 2x=±π/2. And it was shown that tan(2x)=cot(x)=cos(x)=0 when x=π/2. The complete solution is:
x=π/24, 5π/24, π/2 in the given interval.