Question

Solve for $\theta$:

$\tan \theta +cot\theta =2$

Answer 1

Step 1: Use the relation between tangent and cotangent of an angle and reduce given equation into a quadratic equation

Given that

$\tan \theta +cot\theta =2$

We know that $cot\theta =\frac{1}{\tan \theta }$

On substituting $cot\theta =\frac{1}{\tan \theta }$in given equation we get

$\begin{array}{rcl}\tan \theta +\frac{1}{\tan \theta }& =& 2\\ & \Rightarrow & \frac{1+{\tan }^2\theta }{\tan \theta }=2\end{array}$

$\begin{array}{rcl}& \Rightarrow & 1+{\tan }^2\theta =2\tan \theta \\ & & \end{array}$

$\begin{array}{rcl}& \Rightarrow & {\tan }^2\theta -2\tan \theta +1=0\end{array}$

Step 2: Use appropriate algebraic identity

From the identity we get to know that

${\tan }^2\theta -2\tan \theta +1={\left(\tan \theta -1\right)}^2$

$$\begin{gathered} \Rightarrow {\left(\tan \theta -1\right)}^2=0 \\ \Rightarrow \left(\tan \theta -1\right)=0 \\ \Rightarrow \tan \theta =1 \end{gathered}$$

$\Rightarrow \tan \theta =\tan \frac{\mathrm{\pi }}{4}$ $\left[\because \tan \frac{\mathrm{\pi }}{4}=1\right]$

Step 3: Find the values of $\theta$

As we know that if $\tan x=\tan \theta$, then the general solution is $x=n\mathrm{\pi }+\mathrm{\theta }$

$\theta =n\mathrm{\pi }+\frac{\mathrm{\pi }}{4},\mathrm{n}\in \mathrm{Z}$

Hence, $\theta =n\mathrm{\pi }+\frac{\mathrm{\pi }}{4},\mathrm{n}\in \mathrm{Z}$ is the solution of given trigonometric equation.