if α,β,γ are the angles of a triangle, find (1/(tan 𝛂 tan 𝛃))+(1/(tan 𝛃 tan 𝛄))+(1/(tan 𝛄 tan 𝛂))=?


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Question Number 158335 by mr W last updated on 02/Nov/21

$i f α, β, γ a r e t h e a n g l e s o f a t r i a n g l e,$ $f i n d$ $\frac{1}{\tan α \tan β} + \frac{1}{\tan β \tan γ} + \frac{1}{\tan γ \tan α} = ?$
	Answered by puissant last updated on 03/Nov/21

	$α + β + γ = π \to α + β = π - γ;$ $\Rightarrow t a n (α + β) = t a n (π - γ) \Rightarrow \frac{t a n α + t a n β}{1 - t a n α t a n β} = - t a n γ$ $\Rightarrow t a n α + t a n β = - t a n γ + t a n α t a n β t a n γ$ $\Rightarrow t a n α + t a n β + t a n γ = t a n α t a n β t a n γ$ $\frac{1}{t a n α t a n β} + \frac{1}{t a n β t a n γ} + \frac{1}{t a n α t a n γ}$ $= \frac{t a n γ}{t a n α t a n β t a n γ} + \frac{t a n α}{t a n α t a n β t a n γ} + \frac{t a n β}{t a n α t a n β t a n γ}$ $= \frac{t a n α + t a n β + t a n γ}{t a n α t a n β t a n γ} = 1$ $. . . . . . . . . . . L e p u i s s a n t . . . . . . . . . . . .$
	Commented by mr W last updated on 03/Nov/21

	$t h a n k s!$
	Commented by puissant last updated on 03/Nov/21

	${y o u}^{'} r e w e l c o m e s i r$
	Answered by ajfour last updated on 03/Nov/21

	Commented by ajfour last updated on 03/Nov/21

	$r e q u i r e d = \frac{p q}{r^{2}} + \frac{(1 - \frac{p q}{r^{2}})}{(\frac{p}{q} + \frac{q}{r})} (\frac{p}{r} + \frac{q}{r})$ $= 1 .$
	Commented by mr W last updated on 03/Nov/21

	$g r e a t!$
	Commented by ajfour last updated on 03/Nov/21
	yeah, but not worth a like..
	Commented by mr W last updated on 03/Nov/21

	$n o t m a n y p e o p l e c a n c o m e t o t h i s$ $m e t h o d, i t h i n k .$
	Commented by otchereabdullai@gmail.com last updated on 07/Nov/21

	$nice one!$
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