The general solution of equation tan x tan 4x = 1 is (1) (2n + 1)(π/(10)) , n ∈ Z − {n : n = 5k +2; k ∈ Z} (2) (4n − 1)(π/(10)) , n ∈ Z (3) ((nπ)/(10)) , n ∈ Z (4) 2nπ + (π/(10)) , n ∈ Z


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Question Number 14030 by Tinkutara last updated on 27/May/17

$The general solution of equation$ $\tan x \tan 4 x = 1 is$ $(1) (2 n + 1) \frac{π}{10}, n \in Z - {n : n = 5 k + 2; k \in Z}$ $(2) (4 n - 1) \frac{π}{10}, n \in Z$ $(3) \frac{n π}{10}, n \in Z$ $(4) 2 n π + \frac{π}{10}, n \in Z$
	Answered by ajfour last updated on 27/May/17

	$\tan (x + 4 x) = \frac{\tan x + \tan 4 x}{1 - \tan x \tan 4 x}$ $\tan x \tan 4 x = 1 \Rightarrow \frac{1}{\tan 5 x} = 0$ $5 x = n π + \frac{π}{2}$ $x = \frac{n π}{5} + \frac{π}{10} = (2 n + 1) \frac{π}{10}; n \in Z$
	Commented by Tinkutara last updated on 27/May/17

	$Option (1) is right but n \notin Z instead$ $n \in Z - {n : n = 5 k + 2; k \in Z} . Why ?$
	Commented by mrW1 last updated on 27/May/17

	$n o t e v e r y n i s o k .$ $n s h o u l d n o t b e 2, 7, . . .$ $o t h e r w i s e 4 x = 2 π, 4 π, . . .$ $a n d \tan 4 x = 0, t h i s i s n o t a l l o w e d,$ $s i n c e \tan 4 x \times \tan x = 1 \neq 0$
	Commented by Tinkutara last updated on 27/May/17

	$Thanks .$
	Commented by ajfour last updated on 27/May/17

	$v e r y t r u e, a n d a l s o$ $n \neq \frac{(5 m - 2)}{4}, m \in Z$ $s i n c e w i t h t h e d e f i n i t i o n$ $x = \frac{π}{10} + n \frac{π}{5} [o p t i o n (1)]$ $∙ \tan x i s n e v e r = 0$ $∙ \tan 5 x i s n e v e r z e r o a n d a l w a y s$ $n o t d e f i n e d (w h i c h i s r e q u i r e d$ $f o r \tan x \tan 4 x = 1)$ $∙ \tan x i s n o t d e f i n e d f o r$ $t w o l o c a t i o n s o u t o f t e n .$ $∙ \tan 4 x i s z e r o f o r t w o l o c a t i o n s$ $o u t o f f i v e$ $f i r s t a n d s e c o n d a r e g o o d p o i n t s$ $t h i r d a n d f o u r t h n o t s o g o o d .$ $so we should ensure$ $f o r \tan x \neq u n d e f i n e d$ $x = \frac{π}{10} + n \frac{π}{5} \neq k π + \frac{π}{2}$ $\Rightarrow n \neq 5 k + 2, k \in Z$ $a l s o f o r \tan 4 x \neq 0$ $4 x = 4 (\frac{π}{10} + n \frac{π}{5}) \neq m π$ $\Rightarrow \frac{2}{5} + \frac{4 n}{5} \neq m$ $o r n \neq \frac{5 m - 2}{4}, m \in Z .$
	Commented by Tinkutara last updated on 27/May/17

	$Thanks .$
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