# Tan 20 tan 40 tan 80 = tan 60 prove with all steps without a calculator

somi teez
105 Points
7 years ago
Hi,
taking L.H.S ,
tan 40 * tan 80* tan 20=(sin 40 *sin 80) * sin 20 /(cos 40 *cos 80 ) cos 20
now, multiplying numerator and denominator by 2,
=(2*sin 40*sin 80) *sin 20 /(2 cos 40*cos 80) cos 20]
=(cos 40- cos 120)*sin 20/( (cos 120+ cos 40) cos 20
=(2 cos 40 +1) sin 20 /(2 cos 40-1) cos 20
=(2 cos 40*sin 20 +sin 20) /(2 cos 40 cos 20 -cos 20)
=(sin 60 -sin 20 +sin 20)/ (cos 60+sin 20-sin 20)
= sin 60/cos 60
=tan 60.........proved  (Using basic addition ans subtr. of trigno...given in book)
approve it if you agree and understood the answer
BEST OF LUCK!!

Rachit
11 Points
6 years ago
there is a formula tan A . (tan π/3 - A) . (tan π/3 +A ) = tan 3Atherefore [π/3=60]tan 20. (tan (60-20)).(tan (60+20)) = tan 3(20) = tan 60 hence proved
Arvinder Singh
13 Points
5 years ago

tan 40 * tan 80* tan 20=(sin 40 *sin 80) * sin 20 /(cos 40 *cos 80 ) cos 20
now, multiplying numerator and denominator by 2,
=(2*sin 40*sin 80) *sin 20 /(2 cos 40*cos 80) cos 20]
=(cos 40- cos 120)*sin 20/( (cos 120+ cos 40) cos 20
=(2 cos 40 +1) sin 20 /(2 cos 40-1) cos 20
=(2 cos 40*sin 20 +sin 20) /(2 cos 40 cos 20 -cos 20)
=(sin 60 -sin 20 +sin 20)/ (cos 60+sin 20-sin 20)
= sin 60/cos 60
=tan 60.........proved
Rishi Sharma
3 years ago
Hi,
taking L.H.S ,
tan 40 * tan 80* tan 20 = (sin 40 *sin 80) * sin 20 /(cos 40 *cos 80 ) cos 20
now, multiplying numerator and denominator by 2,
= (2*sin 40*sin 80) *sin 20 /(2 cos 40*cos 80) cos 20]
= (cos 40- cos 120)*sin 20/( (cos 120+ cos 40) cos 20
= (2 cos 40 +1) sin 20 /(2 cos 40-1) cos 20
= (2 cos 40*sin 20 +sin 20) /(2 cos 40 cos 20 -cos 20)
= (sin 60 -sin 20 +sin 20)/ (cos 60+sin 20-sin 20)
= sin 60/cos 60
= tan 60 proved
Thanks
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