sin10+2sin10sin20sin40
=sin10+2sin10sin(30-10)sin(30+10)
=sin10+2sin10(sin30cos10=sin10cos30)(sin30cos10+sin10cos30)
=sin10+2sin10(1/4 * ( (cos10)^2 - 3(sin10)^2))
=sin10*(1+1/2(cos10)^2 - 3/2 (sin10)^2)
=sin10 * (1/2(cos10)^2+1/2 (sin10)^2+1/2(cos10)^2 - 3/2 (sin10)^2 +1/2)
=sin10 * ( (cos10)^2-(sin10)^2+1/2)
=sin10 * (cos20 + 1/2)
=sin10 * (sin70 + sin30)
=sin10 * (2sin50cos20)
=2sin10cos20sin50
=2sin10cos10cos20sin50/cos10
=1/2 sin40cos40/cos10
=1/4 sin80/cos10
=1/4