# The value of tan6°tan42°tan66°tan78° is

1

2015-05-05T15:04:06+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
From the question, we see that    66 = 60 + 6 ,      42 = 60 - 18  and  78 = 60 +18

Tan42  Tan 78
= Tan (60 - 18) * Tan (60 + 18)
= (Tan 60 - tan 18) / (1+tan60 tan18)  * (tan60 + tan18) / (1 - Tan60 tan18)
= (√3 - tan18) (√3 + tan18) / [ (1+√3 tan18)(1-√3 tan18) ]
= (3 - tan²18) / (1 - 3 tan²18)      ---- (1)
=  (3 Cos²18 - Sin²18) / (Cos²18 - 3 Sin²18)
=  (3 - 4 Sin²18) / (4 Cos²18 - 3)
=  [ Sin 3*18 / Sin 18 ] / [ Cos 3*18 / Cos 18 ]
= Tan 54 / Tan 18                                       ---- (2)
from the formula Sin 3A = sinA (3 - 4 sin²A)  and Cos3A = CosA (4 Cos²A - 3)

Tan 6
Tan 66
= Tan (60 + 6) * tan 6 * tan (60-6) / tan 54
= tan (60 - 6) * tan (60 +6) * tan 6 / tan 54
= [(√3 + tan 6) / (1 -√3 tan 6) ] * [(√3 + tan 6) / (1 - √3 tan6)] * tan 6 / tan 54
= [ (3 - tan² 6) / (1 - 3 tan² 6) ] * tan 6 / tan 54
= [ (3 Cos² 6 - Sin² 6) / ( Cos² 6 - 3 Sin² 6) ] * [ tan 6 / tan 54 ]
= [ (3 - 4 Sin² 6) / (4 Cos² 6 - 3) ] * tan 6 / tan 54 ]
= [ (3 Sin 6 - 4 Sin³ 6) / (4 Cos³ 6 - 3 Cos 6) ] / tan 54 ]
= Sin (3*6) / Cos (3*6)  / tan 54
= Tan 18 / Tan 54

Hence,  Tan 6 tan 42 tan 66  tan 78
= Tan 54 / Tan 18  *  tan 18 / Tan 54
= 1
================================

Another way:

Tan (60 - A)
= (tan 60 - tan A) / (1 + tan 60 tan A)
= (√3 - Tan A) / (1 + √3 tan A)                    ---- (1)
Tan (60 +A)
= (tan 60 + tan A) / (1 - Tan60 Tan A)
= ( √3 + Tan A) / (1 - √3 Tan A)                  ---- (2)

Tan (60 - A) * Tan (60 + A)
= (3 - Tan²A) / (1 - 3 Tan²A)
= (3 - sec²A + 1) / (1 - 3 sec²A + 3)
= (4 Cos²A -1) / (4 Cos²A - 3)              --- (3)

Sin 3A = 3 Sin A - 4 Sin³ A
Cos 3A = 4 Cos³ A - 3 Cos A
Tan 3A  =  [ 3 Sin A - 4 Sin³A ] / [ 4 Cos³ A - 3 Cos A]
=  Tan A [ 4 Cos² A - 1 ] / [ 4 Cos² A - 3 ]            ---- (4)
= Tan A [ 4 - sec² A ] / [ 4 - 3 Sec² A ]
=   Tan A [ 3 - Tan² A ] / [ 1 - 3 Tan² A ]               --- (5)

From  (1), (2) , (3) , (4)  we have:

Tan 3A / Tan A = Tan (60 - A)  Tan (60 + A)    --- (6)
================
Now,    let  A = 18°  in (6)
=>  Tan 54° / Tan 18° = Tan 42° Tan 78°        ---- (7)

Let A = 6° in (6)
=>  Tan 18°/Tan 6° = Tan 54°  Tan 66°
=>  Tan 18° / Tan 54° = Tan 6°  Tan 66°        --- (8)

Multiplying (8) and (7):
Tan 6°  Tan 42° Tan 66° Tan 78°
=  tan 54 / tan 18   *  tan 18/ Tan 54
= 1

please click on thanks blue button above