Question

tan20+tan40+√3 tan 20 ×tan 40

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Answer 1

it is given that tan20° + tan40° + √3 tan20° × tan40°

now tell me , can we write tan60° = tan(40° + 20°) ? of course yes !

now use the formula, tan(A + B) = (tanA + tanB)/(1 - tanA.tanB)

so, tan(40° + 20°) = (tan40° + tan20°)/(1 - tan40°.tan20°)

now, tan60° = (tan40° + tan20°)/(1 - tan40°.tan20°)

or, √3(1 - tan40°. tan20°) = tan40° + tan20°

or, √3 - √3tan20°. tan40° = tan40° + tan20°

or, √3 = tan20° + tan40° + √3 tan20° . tan40°

hence, value of (tan20° + tan40° + √3tan20°. tan40°) = √3