Question

Prove the following:

tan 20° tan 80° cot 50° = `sqrt(3)`

Answer 1

L.H.S. = tan 20° tan 80° cot 50°

= tan 20° tan 80° cot (90° − 40°)

= tan 20° tan 80° tan 40°

= tan 20° tan (60° + 20°) tan (60° − 20°)

`=tan20°((tan60°+tan20°)/(1-tan60°tan20°))((tan60°-tan20°)/(1+tan60°tan20°))`

= `tan20°((sqrt3+tan20°)/(1-sqrt3tan20°))((sqrt3-tan20°)/(1+sqrt3tan20°))`

= tan 20° `[((sqrt(3))^2-tan^2 20°)/(1^2-(sqrt3tan20°)^2)]`

= tan 20° `((3-tan^2 20°)/(1-3tan^2 20°))`

= `(3tan20°-tan^3 20°)/(1-3tan^2 20°)`

= tan 3 (20°)
= tan 60°
= `sqrt3`
= R.H.S.