Prove that?

#sin 10*sin 30*sin 50*sin 70=1/16#

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2 Answers
Umer F.
Feb 17, 2018

See steps below...

Explanation:

Given: #" "# #Sin 10*Sin 30*Sin*50*sin 70#

Put #\ \ \ # #Sin30=1/2#

#=\ (1/ 2) Sin 10 * Sin 50 * sin 70#

Multiply and divide with #2Cos 10#

#=\ \frac{2\ Cos10\cdot Sin10\cdot Sin\ 50\cdot sin\ 70}{2\cdot 2Cos10}#

Apply the double angle formula #\ \ ##2Sin\ x\ Cos\ x\ =\ Sin\ 2x\ #

#=\ \frac{Sin(2*10)\cdot Sin\ 50\cdot sin\ 70}{4Cos10}#

#=\ \frac{Sin\ 20\cdot Sin\ 50\cdot sin\ 70}{4Cos10}#

Rewrite #\ \ \ # #sin(70)=sin(90-20)=cos(20)# #\ \ \ # because #\ \ \ ##sin(\pi/2-x)=cos(x)#

#=\ \frac{Sin\ 20\cdot Cos\ 20\cdot sin\ 50}{4Cos10}#

Multiply and divide by 2 and apply the double angle formula again:

#=\ \frac{2\cdot Sin\ 20\cdot Cos\ 20\cdot sin\ 50}{2\cdot 4Cos10}#

#=\ \frac{Sin\ 40\cdot sin\ 50}{8Cos10}#

Repeat the same steps, #\ \ \ # #sin(50)=sin(90-40)=cos(40)#

#=\ \frac{2\cdot Sin\ 40\cdot cos\ 40}{2\cdot 8Cos10}#

#=\ \frac{Sin\ 80}{16Cos10}#

Apply #\ \ \ ##cos(10)=cos(90-80)=sin(80)# #\ \ \ # because #\ \ \ ##cos(\pi/2-x)=sin(x)#

#=\ \frac{Sin\ 80}{16sin\ 80}#

Cancel out #sin\ 80#

#=1/16#

That's it!

P dilip_k
Feb 17, 2018

#LHS=sin 10*sin 30*sin 50*sin 70#

#=sin(90-80)1/2sin(90- 40)*sin(90-20)#

#=1/(4sin20)cos80cos40(2sin20cos20)#

#=1/(8sin20)cos80(2cos40sin40)#

#=1/(16 sin20)(2cos80sin80)#

#=1/(16 sin20)sin(180-20)#

#=1/(16 sin20)sin20#

#=1/16=RHS#

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