#LHS=(1 + tan2θtan3θ)/(1+ tanθtan3θ)#
#=(1 + (sin2θsin3θ)/(cos2thetacos3theta))/(1+ (sinθsin3θ)/(costheta cos3theta))#
#=((cos3thetacos2theta + sin2θsin3θ)/(cos2thetacos3theta))/((cos3thetacostheta+ sinθsin3θ)/(costheta cos3theta))#
#=((cos(3theta-2theta))/(cos2thetacos3theta))/((cos(3theta-theta))/(costheta cos3theta))#
#=((costheta)/(cos2thetacos3theta))/((cos(2theta))/(costheta cos3theta))#
#=cos^2θ/cos^2 (2θ)=RHS#