Question

prove that tan72=tan18+2tan54

Answer 1

SINCE,

$= > 72 - 18 = 54$

Take "tan" on both sides :-

$= > tan(72 - 18) = tan54$
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Using trigonometric identity :-

$= > tan( \alpha - \beta ) = \frac{tan \alpha - tan \beta }{1 + tan \alpha .tan \beta }$
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Then,

$= > tan(72 - 18) = tan54 \\ \\ = > \frac{tan72 - tan18}{1 + tan72.tan18} = tan54 \\ \\ = > \frac{tan72 - tan18}{1 + tan(90 - 18).tan18} = tan54 \\ \\ = > \frac{tan72 - tan18}{1 + cot18.tan18} = tan54 \\ \\ = > \frac{tan72 - tan18}{1 + 1} = tan54 \\ \\ = > tan72 - tan18 = 2tan54 \\ \\ = > tan72 = tan18 + 2tan54$
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