Prove that cot 10 cot 30 cot 50 cot 70 = 3.

rewrite as

first note cot 30 = sqrt(3)

rewrite the rest as

cos(10)cos(50)cos(70)     sin(80)sin(40)sin(20)
---------------------- = ---------------------
sin(10)sin(50)sin(70)        cos(80)cos(40)cos(20)

Lets deal with the denominator first
note that sin(160)=sin(20) and apply the half angle sine identity 3 times:
sin(20)=2sin(80)cos(80)
sin(20)=4sin(40)cos(40)cos(80)
sin(20)=8sin(20)cos(20)cos(40)cos(80)
cos(20)cos(40)cos(80) = 1/8

By similar methods the numerator is sqrt(3)/8
so the whole fraction equals sqrt(3)

multiply that by cot(3) and you get your 3.

Posted by Jer on 2013-10-11 16:49:43