See the derivation in the Explanation Section below.

Suppose that ##(30+theta)=alpha & (60-theta)=beta## so that

##alpha+beta=90 or beta=90-alpha##.

Sub.ing these in the given eqn. we have

##tanalpha=2tanbeta=2tan(90-alpha)=2cotalpha=2/tanalpha## or

##tan^2alpha=2##

##:. tan^2(30+theta)=2##

##:. ((tan30+tantheta)/(1-tan30*tantheta))^2=2##.

##:. ((1/sqrt3+t)/(1-1/sqrt3*t))^2=2 where t=tantheta##

##:. (1+sqrt3*t)^2=2(sqrt3-t)^2##.

##:. 1+2sqrt3*t+3t^2=6-4sqrt3*t+2t^2##.

##:. t^2+6sqrt3*t-5=0## i.e.

##tan^2theta+6sqrt3tantheta-5=0## as desired!

Enjoy Maths.!.