Question

Solve the equation 2 tan x - cot x + 1 = 0

Answer 1

2 tan x - cot x + 1 = 0

$\cot(x) = \frac{1}{ \tan(x) }$

put the value of cot X into the equation

$2 \tan(x) - \frac{1}{ \tan(x) } + 1 = 0 \\ \\ \frac{2 \: {tan}^{2}x - 1 + tan \: x }{tan \: x} = 0 \\ \\ 2 \: {tan}^{2}x - 1 + tan \: x = 0 \\ \\ 2 \: {tan}^{2}x + tan \: x - 1 = 0$

factorise the above equation

$2 {tan}^{2} x + 2 \: tanx - tan \: x - 1 = 0 \\ \\ 2 \: tanx(tan \: x + 1) - 1(tan \: x + 1) = 0 \\ \\ (2 \: tan \: x - 1)(tan \: x + 1) = 0 \\ \\ (tan \: x + 1) = 0 \\ \\ tan \: x = - 1 \\ \\ x = {tan}^{ - 1} ( - 1) \\ \\ x = \frac{ - \pi}{4}$

for second value of x use another factor

$(2 \: tan \: x - 1) = 0 \\ \\ 2 \: tan \: x = 1 \\ \\ tan \: x = \frac{1}{2} \\ \\ x = {tan}^{ - 1} ( \frac{1}{2} )$

by checking trigonometry functions table we can calculate the value x.