We take ,
LHS=tan 20^circ+tan80^circ+tan140^circ
color(white)(LHS)=tan20^circ+tan(60^circ+20^circ)+tan(120^circ+20^circ)
color(white)(LHS)=tan20^circ+(tan60^circ+tan20^circ)/(1-tan60^circtan20^circ)+(tan120^circ+tan20^circ)/(1-tan120^circtan20^circ)
Subst. color(blue)(tan60^circ=sqrt3 ,tan120^circ=-sqrt3 and tan20^circ=t
LHS=t+(sqrt3+t)/(1-sqrt3t)+(-sqrt3+t)/(1+sqrt3t)
color(white)(LHS)=t+{(sqrt3+t)(1+sqrt3t)+(-sqrt3+t)(1-sqrt3t))/((1-sqrt3t)(1+sqrt3t))
color(white)(LHS)=t+(sqrt3+3t+t+sqrt3t^2-sqrt3+3t+t-sqrt3t^2)/(1-3t^2)
color(white)(LHS)=t+(8t)/(1-3t^2)
color(white)(LHS)=(t-3t^3+8t)/(1-3t^2)
color(white)(LHS)=(9t-3t^3)/(1-3t^2)
color(white)(LHS)=3[(3t-t^3)/(1-3t^2)]towhere,color(blue)(t=tan20^circ
color(white)(LHS)=3[(3tan20^circ-tan^3 20^circ)/(1-3tan^2 20^circ)]
color(white)(LHS)=3[tan3(20^circ)]toApply(2) for theta=20^circ
LHS=3tan60^circ
LHS=3sqrt3=RHS
Note :
(1) tan(A+B)=(tanA+tanB)/(1-tanAtanB)
(2)tan3theta=(3tantheta-tan^3theta)/(1-3tan^2theta)