Question

The value of tan6*tan42*tan66*tan78 is

Answer 1

This is a Methiod of Doing it .

Use Trigonometric Identity

tanx * tan(60 - x) * tan(60 + x) = tan(3x)

set x = 6

tan6 * tan54 * tan66 = tan18 .................... ((1))

Similarly, set x = 18

tan18 * tan42 * tan78 = tan54 ...................... ((2))

((1)) * ((2)):

tan6 * tan54 * tan66 * tan18 * tan42 * tan78 = tan18 * tan54

tan6 * tan66 * tan42 * tan78 = 1

PROVEN


ANOTHER METHOD

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