Question

Sin6.sin42.sin66.sin78

Answer 1

You might have heard the identity
$\sin( \alpha ) \times \sin(60 - \alpha ) \times \sin(60 + \alpha ) \\ = \frac{ \sin( 3\alpha ) }{4}$
If not, then don't worry it's very easy to prove, you can try it yourself.....
Meanwhile,
$\sin(6) \sin(42) \sin(66) \sin(78) \\ \sin(6) \sin(66) \times \sin(42) \sin(78) \\ \frac{ \sin(6) \sin(54) \sin(66) }{ \sin(54) } \times \frac{ \sin(18) \sin(42) \sin(72) }{ \sin(18) } \\ \frac{ \sin(3 \times 6) }{ 4\sin(54) } \times \frac{ \sin(18 \times 3) }{4 \sin(18) } \\ \frac{ \sin(18) }{4 \sin(54) } \times \frac{ \sin(54) }{4 \sin(18) } \\ = \frac{1}{16}$