Question

Prove :tan 240° +tan 30°
------------------------ =2
tan 240° -tan 30°
​

Answer 1

To prove --->

tan 240° + tan 30°

------------------------------ = 2

tan 240° - tan 30°

Proof ---> We know that in third quadrant tan is positive i e

tan ( 180° + θ ) = tanθ

So tan240° = tan ( 180° + 60° )

angle of 240° lies in third quadrant so

= tan 60°

tan 240° = √3

Now,

tan 240° - tan 30° = √3 - 1 / √3

Taking √3 as LCM we get

= (√3 √3 - 1 )/ √3

= 3 - 1 / √3

= 2 / √3

Now

tan 240° + tan 30° = √3 + 1 / √3

= (√3 √3 + 1) / √3

= ( 3 + 1 )/ √3

= 4 / √3

Now

LHS

=( tan240°+tan30°) / (tan240° - tan 30°)

Putting ( tan 240° + tan 30° ) = 4 / √3

and ( tan 240° - tan 30° ) = 2 / √3

= ( 4 / √3 ) / ( 2 / √3 )

= 4 √3 / 2 √3

√3 is cancel out from numerator and denominator we get

= 4 / 2

= 2 = RHS