Question

tan6tan42tan66tan78=1​

Answer 1

Answer:

tan 6 tan 42 tan 66 tan 78 = 1​   [ Proved ]

Step-by-step explanation:

Given ;

tan 6 tan 42 tan 66 tan 78 = 1​

L.H.S. = tan 6 tan 42 tan 66 tan 78

= tan 66 tan 6 tan 78 tan 42

Convert into sin / cos term

= sin 66 sin 6 sin 78 sin 42 / cos 66 cos 6 cos 78 cos 42

Multiply and divide by 2

= [ (2 sin 66 sin 6 ) ( 2 sin 78 sin 42 )  / ( 2 cos 66 cos 6 )  ( 2 cos 78 cos 42 ) ]

Using 2 sin A sin B = cos ( A - B ) - cos ( A + B )

And  2 cos A cos B = cos ( A - B ) + cos ( A + B )

= [ ( cos 60 - cos 72 ) ( cos 36 - cos 120 ) / ( cos 60 + cos 72 ) ( cos 36 + cos 120 ) ]

Using complimentary formula cos ( 90 - A ) = sin A , cos ( 90 + A ) = - sin A

= [ ( cos 60 - sin 18 ) ( cos 36 + sin 30 ) / ( cos 60 + sin 18 ) ( cos 36 - sin 30 ) ]

We have value of sin 18 = √ 5 - 1 / 4, sin 30 = 1 / 2 and cos 36 = √ 5 + 1 / 4

= [ ( 1 / 2 - √ 5 - 1 / 4 ) ( √ 5 + 1 / 4 + 1 / 2 ) / ( 1 / 2 + √ 5 - 1 / 4 ) ( √ 5 + 1 / 4 - 1 / 2 ) ]

Using ( a - b ) ( a + b ) = a² - b² in numerator

= ( 3 - √ 5 ) ( 3 + √ 5 ) / ( √ 5 - 1 ) ( √ 5 + 1 )

= 9 - 5 / 5 - 1

= 4 / 4

= 1

L.H.S. = R.H.S.

Hence proved .