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

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

Grade:11

4 Answers

somi teez
105 Points
6 years ago
Hi,
i will try to help you 
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
4 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
askIITians Faculty 646 Points
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
I hope above solution will clear your doubts.
Please feel free to post as many doubts on our discussion forum as you can.
All the best.

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