The-general-solution-of-equation-tan-x-tan-4x-1-is-1-2n-1-pi-10-n-Z-n-n-5k-2-k-Z-2-4n-1-pi-10-n-Z-3-npi-10-n-Z-4-2npi-pi-10-n-Z-

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, \dots

o t h e r w i s e 4 x = 2 π, 4 π, \dots

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 .