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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

\dots \dots \dots . . L e p u i s s a n t \dots \dots \dots \dots

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!