2sin50°+sin10°[1+√3tan10°)√(1+cos20°)
=[2sin50°+sin10°*(cos10°+√3sin10°)/cos10°]*√2cos10°
=2√2sin50°cos10°+√2sin10°cos10°+√6(sin10°)^2
=2√2sin50°cos10°+√2/2sin20°+√6/2-√6/2cos20°
=2√2sin50°cos10°+√2(1/2*sin20°-√3/2*cos20°)+√6/2
=2√2sin50°cos10°-√2cos50°+√6/2
=2√2cos40cos10°-√2cos(40°+10°)+√6/2
=2√2cos40°cos10°-√2cos40°cos10°
+√2sin40°sin10°+√6/2
=√2(cos40°cos10°+sin40°sin10°)+√6/2
=√2cos(40°-10°)+√6/2
=√2cos30°+√6/2
=√6/2+√6/2
=√6