If-tan-x-tan-pi-3-x-tan-2pi-3-x-3-a-tan-x-1-b-tan-2x-1-c-tan-3x-1-d-None-of-above-

Question Number 16681 by rish@bh last updated on 25/Jun/17

If

\tan x + \tan (\frac{π}{3} + x) + \tan (\frac{2 π}{3} + x) = 3

a) \tan x = 1

b) \tan 2 x = 1

c) \tan 3 x = 1

d) None of above

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/Jun/17

\frac{\sqrt{3} + t g x}{1 - \sqrt{3} t g x} + \frac{- \sqrt{3} + t g x}{1 + \sqrt{3} t g x} = 3 - t g x \Rightarrow

\frac{\sqrt{3} + 3 t g x + t g x + \sqrt{3} {t g}^{2} x - \sqrt{3} + t g x + 3 t g x - \sqrt{3} {t g}^{2} x}{1 - 3 {t g}^{2} x} = 3 - t g x

\Rightarrow \frac{8 t g x}{1 - 3 {t g}^{2} x} = 3 - t g x \Rightarrow 8 t g x = 3 - t g x - 9 {t g}^{2} x + 3 {t g}^{3} x

\Rightarrow {t g}^{3} x - 3 {t g}^{2} x - 3 t g x + 1 = 0

\Rightarrow (t g x + 1) ({t g}^{2} x - 4 t g x + 1) = 0

\Rightarrow t g x = - 1, t g x = \frac{4 \pm \sqrt{16 - 4}}{2} = 2 \pm \sqrt{3} .

t g x = - 1 \Rightarrow t g 2 x = \infty, t g 3 x = 1

t g x = 2 - \sqrt{3} \Rightarrow x = {\begin{cases} \frac{π}{12} \\ t g 2 x = \frac{\sqrt{3}}{3}, t g 3 x = 1 \end{cases}

t g x = 2 + \sqrt{3} \Rightarrow x = {\begin{cases} \frac{5 π}{12} \\ t g 2 x = - \frac{\sqrt{3}}{3}, t g 3 x = 1 \end{cases}

s o i n a l l c a s e s w e h a v e : t g 3 x = 1 \Rightarrow a n s w e r (c) .

w a y : 2

x + (x + \frac{π}{3}) + (x + \frac{2 π}{3}) = π + 3 x

\Rightarrow t g x + t g (x + \frac{π}{3}) + t g (x + \frac{2 π}{3}) = t g x . t g (x + \frac{π}{3}) t g (x + \frac{2 π}{3})

t g (x + \frac{2 π}{3}) = - t g (π - (\frac{2 π}{3} + x)) = - t g (\frac{π}{3} - x) =

= t g (x - \frac{π}{3})

t g 3 x = t g x . t g (x + \frac{π}{3}) t g (x - \frac{π}{3})

\Rightarrow 3 t g 3 x = 3 \Rightarrow t g 3 x = 1 \Rightarrow a n s w e r (c) .