Prove that Sin 10 degree sin 30 degree sin 50 degree sin 70 degree = 1/16?

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Answer 1 · 2018-06-02T13:33:55

$LHS=sin10sin30sin50sin70$

$=cos(90-10)sin30cos(90-50)cos(90-70)$

$=cos(80)*1/2*cos(40)cos(20)$
$=1/(4 sin20)cos(80)cos(40)*2sin20cos(20)$

$=1/(8 sin20)cos(80)*2cos(40)sin(40)$

$=1/(16 sin20)*2cos(80)sin(80)$

$=1/(16 sin20)*sin(160)$

$=1/(16 sin20)*sin(180-20)$

$=1/(16 sin20)*sin(20)=1/16=RHS$

Answer 2 · 2018-06-02T14:50:13

As, $sinxsin(60+x)sin(60-x)=1/4sin3x$

$rarrsin10sin30sin50sin70$

$=1/2[sin10sin(60+10)sin(60-10)]$

$=1/2[1/4sin(3xx10)]=1/8sin30=1/8*1/2=1/16$