solve ( x,y ∈ R ) { (( tan(x ) + tan (y )=2)),(( tan(2x ) + tan( 2y ) = 2)) :} −−−−−−−−


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Question Number 176490 by mnjuly1970 last updated on 20/Sep/22

$s o l v e (x, y \in R)$ ${\begin{cases} \tan (x) + \tan (y) = 2 \\ \tan (2 x) + \tan (2 y) = 2 \end{cases}$ $- - - - - - - -$
	Answered by ajfour last updated on 20/Sep/22

	$\frac{2 \tan x}{1 - \tan^{2} x} + \frac{2 \tan y}{1 - \tan^{2} y} = 2$ $\Rightarrow \frac{\tan x (1 - \tan^{2} y) + \tan y (1 - \tan^{2} x)}{1 + {(\tan x \tan y)}^{2} - \tan^{2} x - \tan^{2} y} = 1$ $\Rightarrow \frac{2 - 2 \tan x \tan y}{1 - 4 + 2 \tan x \tan y + {(\tan x \tan y)}^{2}} = 1$ $c a l l \tan x = p, \tan y = q, & p q = m$ $w e t h e n h a v e p + q = 2 &$ $2 - 2 m = 1 + 2 m + m^{2}$ $\Rightarrow m^{2} + 4 m + 4 = 5$ $m = \pm \sqrt{5} - 2$ $p, q = 1 \pm \sqrt{1 + 2 \mp \sqrt{5}}$ $c h o o s i n g - s i g n$ $p, q = 1 \pm \sqrt{3 - \sqrt{5}}$ $w h i l e i f w e c h o o s e + s i g n$ $p, q = 1 \pm \sqrt{3 + \sqrt{5}}$ $A n d w h e n$ $p = \tan x$ $x = \tan^{- 1} p + m π \forall m \in Z$
	Commented by Tawa11 last updated on 20/Sep/22

	$I have subscribe too .$
	Commented by Tawa11 last updated on 20/Sep/22

	$Great sir$
	Commented by ajfour last updated on 20/Sep/22
	https://youtu.be/GmIVICE0YrQ
	Commented by mnjuly1970 last updated on 20/Sep/22

	$t h a n k s a l o t s i r a j f o r$ $y o u r y o u t u b e i s v e r y v e r y$ $e x c e l l e n t . . . g r a t e f u l s i r . . .$
	Commented by ajfour last updated on 20/Sep/22

	$t h a n k s f o r v i e w i n g, a p p r e c i a t i n g!$
	Answered by behi834171 last updated on 20/Sep/22
	$tgx=m,tgy=n tg(x+y)=(2/(1−mn)) tg(2x+2y)=(2/(1−((2m)/(1−m^2 )).((2n)/(1−n^2 )))) tg(2x+2y)=((2×(2/(1−mn)))/(1−(4/((1−mn)^2 ))))=((4(1−mn))/((1−mn)^2 −4)) ⇒((1−m^2 −n^2 +m^2 n^2 )/(1−n^2 −m^2 +m^2 n^2 −4mn))=((2−2mn)/(m^2 n^2 −2mn−3))⇒ m^2 n^2 −2mn−3−m^4 n^2 +2m^3 n+3m^2 −m^2 n^4 +2mn^3 +3n^2 +m^4 n^4 −2m^3 n^3 −3m^2 n^2 = 2−2n^2 −2m^2 +2m^2 n^2 −8mn−2mn+ 4mn^3 +4m^3 n−2m^3 n^3 +8mn⇒ −2m^2 n^2 −2mn−3−m^4 n^2 +2m^3 n+3m^2 −m^2 n^4 +2mn^3 +3n^2 +m^4 n^4 −2m^3 n^3 −3m^2 n^2 = 2−2n^2 −2m^2 +2m^2 n^2 −2mn+4mn^3 + 4m^3 n−2m^3 n^3 ⇒ −4m^2 n^2 −5−m^4 n^2 −2m^3 n+5m^2 −m^2 n^4 −2mn^3 +5m^2 +m^4 n^4 =0⇒ m^4 n^4 −2m^3 n−2mn^3 −m^4 n^2 −m^2 n^4 −4m^2 n^2 +5m^2 +5n^2 −5=0 ⇒t^4 −3t(m^2 +n^2 )−t^2 (m^2 +n^2 )−4t^2 + +5(m^2 +n^2 )−5=0 ⇒t^4 −3t(4−2t)−t^2 (4−2t)−4t^2 + +5(4−2t)−5=0⇒ t^4 −12t+6t^2 −4t^2 +2t^3 −4t^2 +20−10t−5=0 ⇒t^4 +2t^3 −2t^2 −22t+15=0⇒ t=mn=0.67, t=mn=2.16 ⇒ { ((m+n=2)),((mn=0.67 or 2.16)) :} ⇒z^2 −2z+[0.67 or 2.16]=0 ⇒z−1=±0.57 ⇒ { ((tgx=1.57 or 0.43)),((tgy=0.43 or 1.57)) :}$
	$t g x = m, t g y = n$ $t g (x + y) = \frac{2}{1 - m n}$ $t g (2 x + 2 y) = \frac{2}{1 - \frac{2 m}{1 - m^{2}} . \frac{2 n}{1 - n^{2}}}$ $t g (2 x + 2 y) = \frac{2 \times \frac{2}{1 - m n}}{1 - \frac{4}{{(1 - m n)}^{2}}} = \frac{4 (1 - m n)}{{(1 - m n)}^{2} - 4}$ $\Rightarrow \frac{1 - m^{2} - n^{2} + m^{2} n^{2}}{1 - n^{2} - m^{2} + m^{2} n^{2} - 4 m n} = \frac{2 - 2 m n}{m^{2} n^{2} - 2 m n - 3} \Rightarrow$ $m^{2} n^{2} - 2 m n - 3 - m^{4} n^{2} + 2 m^{3} n + 3 m^{2}$ $- m^{2} n^{4} + 2 {m n}^{3} + 3 n^{2} + m^{4} n^{4} - 2 m^{3} n^{3} - 3 m^{2} n^{2} =$ $2 - 2 n^{2} - 2 m^{2} + 2 m^{2} n^{2} - 8 m n - 2 m n +$ $4 {m n}^{3} + 4 m^{3} n - 2 m^{3} n^{3} + 8 m n \Rightarrow$ $- 2 m^{2} n^{2} - 2 m n - 3 - m^{4} n^{2} + 2 m^{3} n + 3 m^{2}$ $- m^{2} n^{4} + 2 {m n}^{3} + 3 n^{2} + m^{4} n^{4} - 2 m^{3} n^{3} - 3 m^{2} n^{2} =$ $2 - 2 n^{2} - 2 m^{2} + 2 m^{2} n^{2} - 2 m n + 4 {m n}^{3} +$ $4 m^{3} n - 2 m^{3} n^{3} \Rightarrow$ $- 4 m^{2} n^{2} - 5 - m^{4} n^{2} - 2 m^{3} n + 5 m^{2} - m^{2} n^{4}$ $- 2 {m n}^{3} + 5 m^{2} + m^{4} n^{4} = 0 \Rightarrow$ $m^{4} n^{4} - 2 m^{3} n - 2 {m n}^{3} - m^{4} n^{2} - m^{2} n^{4}$ $- 4 m^{2} n^{2} + 5 m^{2} + 5 n^{2} - 5 = 0$ $\Rightarrow t^{4} - 3 t (m^{2} + n^{2}) - t^{2} (m^{2} + n^{2}) - 4 t^{2} +$ $+ 5 (m^{2} + n^{2}) - 5 = 0$ $\Rightarrow t^{4} - 3 t (4 - 2 t) - t^{2} (4 - 2 t) - 4 t^{2} +$ $+ 5 (4 - 2 t) - 5 = 0 \Rightarrow$ $t^{4} - 12 t + 6 t^{2} - 4 t^{2} + 2 t^{3} - 4 t^{2} + 20 - 10 t - 5 = 0$ $\Rightarrow t^{4} + 2 t^{3} - 2 t^{2} - 22 t + 15 = 0 \Rightarrow$ $t = m n = 0.67, t = m n = 2.16$ $\Rightarrow {\begin{cases} m + n = 2 \\ m n = 0.67 o r 2.16 \end{cases}$ $\Rightarrow z^{2} - 2 z + [0.67 o r 2.16] = 0$ $\Rightarrow z - 1 = \pm 0.57$ $\Rightarrow {\begin{cases} t g x = 1.57 o r 0.43 \\ t g y = 0.43 o r 1.57 \end{cases}$
	Commented by Tawa11 last updated on 22/Sep/22

	$Great sir .$
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