-sin(10)sin(60)sin(70)=-0.1413 approx but √3/16=0.1083 approx, so the equation is wrong. In fact, since the three sines are all positive their product is also positive, but the leading negative sign makes it negative, while √3/16 is a positive number, so the equation must be wrong. The expression on the left of the equals sign has to contain one negative factor to counteract the leading negative sign for the result to be positive. All the trig functions for angles up to 90° are positive, so we need one angle or all three angles to be greater than 90° in the expression. Also remember that the sine of all angles up to 180° are positive so we need cosines (or secants) or tangents (or cotangents) in the expression or angles greater than 180° for the sine (or cosecant) to be negative.
sin(70)=sin(60+10)=sin(60)cos(10)+cos(60)sin(10).
-sin(10)sin(60)(sin(60)cos(10)+cos(60)sin(10))=
-sin(10)cos(10)sin2(60)-sin2(10)sin(60)cos(60)=
-sin(20)sin2(60)/2-sin2(10)sin(120)/2.
sin(120)=sin(60), so -sin(10)sin(60)sin(70)=-sin(20)sin2(60)/2-sin2(10)sin(60)/2=
-(sin(60)/2)(sin(20)sin(60)+sin2(10)).
cos(20)=1-2sin2(10), so sin2(10)=(1-cos(20))/2, so:
-sin(10)sin(60)sin(70)=-(sin(60)/2)(sin(20)sin(60)+(1-cos(20))/2).
sin(60)=sin(120)=√3/2, so:
-sin(10)sin(60)sin(70)=-(√3/4)(sin(20)√3/2+(1-cos(20))/2)=-⅜sin(20)-√3/8+(√3/8)cos(20).