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The value of sin 78°-sin 66°-sin 42°+sin 6° is

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(sin 78° - sin 42°) - (sin 66° + sin 6°)
By applying sin rule:
2 cos60° sin18° - 2 cos36° sin30°
2 x 1/2 sin18° - 2 cos 36 x 1/2 
sin 18° - cos 36°
sin 18° - sin 54° 
Again,
- ( sin 54° - sin 18°)
- 2cos(36) sin(18)
Also, we know 
sin(36) = 2 sin(18)cos(18)
sin(18)=sin(36)/(2cos(18))
thus,
sin(54)−sin(18) = cos(36) sin(36) / cos(18) = cos(36) sin(36) / sin (72)
sin(54)−sin(18)= cos(36) sin(36) / 2 sin(36) cos(36) = 1/2
- ( sin 54° - sin 18°) = - 1/2

sin(72

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