Question

Tan 40+tan20+√3tan20•tan40=?​

Answer 1

Step-by-step explanation:

we have 20+ 40 = 60

so tan 60 = tan (20+40)

= (tan 20 + tan 40)/(1- tan 20 tan 40)

or tan 60 - tan 60 tan 20 tan 40 = tan 20 + tan 40

or √3 - √3 tan 20 tan 40 = tan 20 + tan 40

or tan 20 + tan 40 + √3 tan 20 tan 40 = √3

Answer 2

tan(x+y) = (tanx + tany)/(1 - tanxtany)

tan(40+20) = (tan40 + tan20)/(1 - tan40tan20)

cross multiply

tan40 + tan 20 = tan(40+20).(1 - tan40tan20)

tan40 + tan 20 = tan60 (1- tan40tan20)

our question

tan30 + tan20 + √3tan20.tan40

replace

tan60 (1- tan40tan20) + √3tan20.tan40

tan60 - tan60.tan40.tan20 + √3tan20.tan40

we know,  tan60 = √3

√3 - √3.tan40.tan20 + √3.tan40.tan20

= √3

answer is √3