.
#cos80^@-cos40^@+sqrt3cos70^@=?#
#cosA-cosB=-2sin((A+B)/2)sin((A-B)/2)#
#cos80^@-cos40^@=-2sin((80^@+40^@)/2)sin((80^@-40^@)/2)=-2sin60^@sin20^@=-2(sqrt3)/2sin20^@=-sqrt3sin20^@#
Let's plug this in:
#cos80^@-cos40^@+sqrt3cos70^@=-sqrt3sin20^@+sqrt3cos70^@=sqrt3(sin20^@-cos70^@)#
#costheta=sin(pi/2-theta)#
#cos70^@=sin(90^@-70^@)=sin20^@#
Let's plug this in:
#cos80^@-cos40^@+sqrt3cos70^@=sqrt3(sin20^@-cos70^@)=sqrt3(sin20^@-sin20^@)=0#