Question

Find the general solution of the following equation: $2\tan x-cotx+1=0.$

Answer 1

Find the general solution of the given expression

Given: $2\tan x-cotx+1=0.$

$$\begin{gathered} \Rightarrow 2\tan \left(x\right)-\frac{1}{\tan \left(x\right)}+1=0 \\ \Rightarrow 2{\tan }^2\left(x\right)+\tan \left(x\right)-1=0 \\ \Rightarrow 2{\tan }^2\left(x\right)+2\tan \left(x\right)-\tan \left(x\right)-1=0 \\ \Rightarrow 2\tan \left(x\right)\left(\tan \left(x\right)+1\right)-1\left(\tan \left(x\right)+1\right)=0 \\ \Rightarrow \left(2\tan \left(x\right)-1\right)\left(\tan \left(x\right)+1\right)=0 \\ \Rightarrow 2\tan \left(x\right)-1=0and\tan \left(x\right)+1=0 \\ \Rightarrow \tan \left(x\right)=\frac{1}{2},\tan \left(x\right)=-1 \\ \Rightarrow x={\tan }^{-1}\left(\frac{1}{2}\right),x={\tan }^{-1}\left(-1\right) \\ \Rightarrow \theta =n\mathrm{\pi }+\mathrm{\alpha },\mathrm{n}\in \mathrm{I} \\ \Rightarrow x=n\mathrm{\pi }+{\tan }^{-1}\left(\frac{1}{2}\right),\mathrm{x}=\mathrm{n\pi }+\frac{3\mathrm{\pi }}{4} \end{gathered}$$

Hence, the general solution is $x=\mathrm{n\pi }+{\tan }^{-1}\left(\frac{1}{2}\right)\mathrm{or}\mathrm{x}=\mathrm{n\pi }+\frac{3\mathrm{\pi }}{4}$ where$\mathrm{n}\in \mathrm{I}$.