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From the question, we see that 66 = 60 + 6 , 42 = 60 - 18 and 78 = 60 +18

= 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 Cos²18 - Sin²18) / (Cos²18 - 3 Sin²18)

= (3 - 4 Sin²18) / (4 Cos²18 - 3)

= [ Sin 3*18 / Sin 18 ] / [ Cos 3*18 / Cos 18 ]

=

= [(√3 + tan 6) / (1 -√3 tan 6) ] * [(√3 + tan 6) / (1 - √3 tan6)] * 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

=

Hence, Tan 6 tan 42 tan 66 tan 78

= Tan 54 / Tan 18 * tan 18 / Tan 54

================================

= (tan 60 - tan A) / (1 + tan 60 tan A)

=

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

=

= (3 - Tan²A) / (1 - 3 Tan²A)

= (3 - sec²A + 1) / (1 - 3 sec²A + 3)

=

= Tan A [ 4 Cos² A - 1 ] / [ 4 Cos² A - 3 ] ---- (4)

= Tan A [ 4 - sec² A ] / [ 4 - 3 Sec² A ]

=

From (1), (2) , (3) , (4) we have:

================

Now, let A = 18° in (6)

=>

Let A = 6° in (6)

=> Tan 18°/Tan 6° = Tan 54° Tan 66°

=>

Multiplying (8) and (7):

**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 ]

=

**---- (2)***Tan 54 / Tan 18***from the formula***Sin 3A = sinA (3 - 4 sin²A) and Cos3A = CosA (4 Cos²A - 3)*

Tan 6Tan 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:Another way:

*Tan (60 - A)*= (tan 60 - tan A) / (1 + tan 60 tan A)

=

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

=

**---- (2)***( √3 + Tan A) / (1 - √3 Tan A)**T**an (60 - A) * Tan (60 + A)*= (3 - Tan²A) / (1 - 3 Tan²A)

= (3 - sec²A + 1) / (1 - 3 sec²A + 3)

=

**--- (3)***(4 Cos²A -1) / (4 Cos²A - 3)**Sin 3A = 3 Sin A - 4 Sin³ A**Cos 3A = 4 Cos³ A - 3 Cos A***= [ 3 Sin A - 4 Sin³A ] / [ 4 Cos³ A - 3 Cos A]***Tan 3A*= Tan A [ 4 Cos² A - 1 ] / [ 4 Cos² A - 3 ] ---- (4)

= Tan A [ 4 - sec² A ] / [ 4 - 3 Sec² A ]

=

**--- (5)***Tan A [ 3 - Tan² A ] / [ 1 - 3 Tan² A ]*From (1), (2) , (3) , (4) we have:

**--- (6)***Tan 3A / Tan A = Tan (60 - A) Tan (60 + A)*================

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*