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Question
Prove the following:
tan 20° tan 80° cot 50° = `sqrt(3)`
Solution
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.
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