prove that tan20+4sin20=root 3

tan20+4sin20
= sin 20/ cos 20 + 4 sin 20
= ( sin 20 + 4 sin 20 cos 20) / cos 20
= (sin 20 + 2 sin 40)/ cos 20
= (sin 20 + 2 sin (60-20))/ cos 20
= ( sin 20 + 2 sin 60 cos 20 - 2 cos 60 sin 20)/ cos 20
= (sin 20+ 2 sin 60 cos 20 - sin 20)/ cos 20
= 2 sin 60cos 20/ cos 20
= 2 sin 60
= 2(√3/2) = √3

alternatively


we can proceed from (sin 20 + 2 sin 40)/ cos 20

as ( sin 20 + sin 40 + sin 40) / cos 20
= (2 sin 30 cos 10 + sin 40)/ cos 20
= (cos 10 + sin 40)/ cos 20
= ( sin 80 + sin 40)/ cos 20
= 2 sin 60 cos 20/ cos 20
= 2 sin 60
= √3