\sin 50° (1+\sqrt{3}\tan 10° )

=2\cos 40\degree (\dfrac{1}{2} +\dfrac{\sqrt{3}}{2} \dfrac{\sin 10\degree }{\cos 10\degree } )

=\dfrac{2\cos 40\degree }{\cos 10\degree } (\sin 30 \degree \cos 10\degree +\cos 30\degree \sin 10\degree )

=\dfrac{2\cos 40\degree }{\cos 10\degree }\sin (30\degree +10\degree )

=\sin 80\degree /\cos 10\degree

=1