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#