If-and-are-the-solution-of-equation-a-tan-b-sec-c-find-the-value-of-tan-

Question Number 105759 by bramlex last updated on 31/Jul/20

I f α a n d β a r e t h e s o l u t i o n o f

e q u a t i o n a \tan θ + b \sec θ = c .

f i n d t h e v a l u e o f \tan (α + β) .

Commented by bramlex last updated on 31/Jul/20

t h x b o t h

Answered by john santu last updated on 31/Jul/20

\Rightarrow a \tan θ + b \sec θ = c

\Rightarrow a \sin θ + b = c \cos θ

u s i n g i d e n t i t y \sin θ = \frac{2 \tan (\frac{θ}{2})}{1 + \tan^{2} (\frac{θ}{2})}

\cos θ = \frac{1 - \tan^{2} (\frac{θ}{2})}{1 + \tan^{2} (\frac{θ}{2})}

w e h a v e (b + c) \tan^{2} (\frac{θ}{2}) + 2 a \tan (\frac{θ}{2}) + b - c = 0

t h e e q u a t i o n i s q u a d r a t i c i n

\tan (\frac{θ}{2}) a n d t h e n \tan (\frac{α}{2}) & \tan (\frac{β}{2})

a r e t h e r o o t s o f t h i s e q u a t i o n .

b y {V i e t a}^{'} s r u l e

{\begin{cases} \tan (\frac{α}{2}) + \tan (\frac{β}{2}) = - \frac{2 a}{b + c} \\ \tan (\frac{α}{2}) . \tan (\frac{β}{2}) = \frac{b - c}{b + c} \end{cases}

a p p l i e d i d e n t i t y \tan (\frac{α + β}{2}) = \frac{- \frac{2 a}{b + c}}{1 - \frac{b - c}{b + c}}

\tan (\frac{α + β}{2}) = - \frac{a}{c}

u s i n g d o u b l e a n g l e f o r m u l a

\tan (α + β) = \frac{2 (- \frac{a}{c})}{1 - \frac{a^{2}}{c^{2}}} = \frac{2 a c}{a^{2} - c^{2}}

♠ ⧫

Commented by Coronavirus last updated on 01/Aug/20

clear thanks you sir

Answered by bobhans last updated on 31/Jul/20

a \tan θ + b \sec θ = c

\to a \tan θ - c = - b \sec θ

s q u a r i n g b o t h s i d e

{(a \tan θ - c)}^{2} = b^{2} (1 + \tan^{2} θ)

(a^{2} - b^{2}) \tan^{2} θ - 2 a c \tan θ + c^{2} - b^{2} = 0

h a s t h e r o o t s a r e \tan α a n d \tan β

{V i e t a}^{'} s r u l e {\begin{cases} \tan α + \tan β = \frac{2 a c}{a^{2} - b^{2}} \\ \tan α . \tan β = \frac{c^{2} - b^{2}}{a^{2} - b^{2}} \end{cases}

t h e r e f o r e \tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α . \tan β}

= \frac{2 a c}{a^{2} - c^{2}} ★

Commented by Coronavirus last updated on 01/Aug/20

wouah very fast method