Question

How do you solve $\tan x - 2 - 3\cot x = 0$?

Answer 1

We transform the cotangent value to tangent value and form the quadratic equation in terms of tangent. Assume tangent of x as a variable as solve for the variable using factorization method. Substitute back the value of the variable and take inverse tangent function to calculate the value of x.

• $\cot \theta = \dfrac{1}{{\tan \theta }}$
• Factorization method: If ‘p’ and ‘q’ are the roots of a quadratic equation, then we can say the quadratic equation is $(x - p)(x - q) = 0$

Complete step-by-step answer:
We are given the equation $\tan x - 2 - 3\cot x = 0$
Substitute the value of $\cot x = \dfrac{1}{{\tan x}}$in the equation
$\Rightarrow \tan x - 2 - \dfrac{3}{{\tan x}} = 0$
Take LCM on left hand side of the equation
$\Rightarrow \dfrac{{{{\tan }^2}x - 2\tan x - 3}}{{\tan x}} = 0$
Cross multiply the value from denominator of left hand side of the equation to right hand side of the equation
$\Rightarrow {\tan ^2}x - 2\tan x - 3 = 0$
Now we can see this is a quadratic equation in terms of tangent of x
Substitute $\tan x = y$
$\Rightarrow {y^2} - 2y - 3 = 0$
We can write the equation such that the coefficient of x is broken in such a way that its product equals product of coefficient of other two terms and sum equals coefficient of x.
$\Rightarrow {y^2} + y - 3y - 3 = 0$
Take y common from first two terms and -3 common from last two terms
$\Rightarrow y(y + 1) - 3(y + 1) = 0$
Collect the factors
$\Rightarrow (y + 1)(y - 3) = 0$
Equate the factors to 0
$\Rightarrow y + 1 = 0$ and $y - 3 = 0$
Shift constant values to right hand side
$\Rightarrow y = - 1$ and $y = 3$
Now substitute the value of y back i.e. put $y = \tan x$
$\Rightarrow \tan x = - 1$ and $\tan x = 3$
Take inverse trigonometric function on both sides of the equation
$\Rightarrow {\tan ^{ - 1}}\left( {\tan x} \right) = {\tan ^{ - 1}}( - 1)$ and ${\tan ^{ - 1}}\left( {\tan x} \right) = {\tan ^{ - 1}}(3)$
Cancel tangent from inverse tangent function
$\Rightarrow x = {\tan ^{ - 1}}( - 1)$ and $x = {\tan ^{ - 1}}(3)$

$\therefore$Solution of the equation $\tan x - 2 - 3\cot x = 0$ are $x = {\tan ^{ - 1}}( - 1)$and $x = {\tan ^{ - 1}}(3)$.

Note:
Many students get confused while solving for the value of variables in the equation and write the value of $\tan x$ as the answer because the equation is formed in $\tan x$. Keep in mind the variable here is x, so we will calculate the value of x.