ice_syncer, this method is SIMPLIER.

Prove: sin10.sin30.sin50.sin70 = 1/16.

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sin 30 = cos (90 - 30) = cos 60 = 1/2,

sin 10 = cos (90 - 10) = cos 80,

sin 50 = cos (90 - 50) = cos 40,

sin 70 = cos (90 - 70) = cos 20,

sin 160 = sin (180 - 160) = sin 20.

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Let P = (sin 10 sin 50 sin 70)(sin 30) = (cos 20 cos 40 cos 80)(sin 30)

multiply BS by sin 20,

P(sin 20) = (sin 20)(cos 20 cos 40 cos 80)(sin 30)

use the identity: 2 sin A cos A = sin 2A,

P(sin 20) = (sin 20 cos 20)(cos 40 cos 80)(sin 30)

P(sin 20) = (1/2)(sin 40)(cos 40 cos 80)(sin 30)

P(sin 20) = (1/2)(sin 40cos 40)(cos 80)(sin 30)

P(sin 20) = (1/2)(1/2)(sin 80)(cos 80)(sin 30)

P (sin 20) = (1/2)(1/2)(1/2)(sin 160)(sin 30)

P (sin 20) = (1/2)(1/2)(1/2)(sin 20)(sin 30)

P (sin 20) = (1/2)(1/2))1/2)(1/2)(sin 20)

you may cancel sin 20, since sin 20 is not equal to 0.

P = sin10 sin30 sin50 sin70 = 1/16

If you multiply by BS by sin 40, it will do the same trick, likewise with sin 80. try it . . . .