Question

How to find the value of $\cot \left(4x\right)$

Answer 1

Find the value of the given trigonometric function

$$\begin{gathered} \cot \left(4x\right)=\frac{1}{\tan \left(4\mathrm{x}\right)} \\ \Rightarrow \cot \left(4\mathrm{x}\right)=\frac{1-{\tan }^2\left(2\mathrm{x}\right)}{2\tan \left(2\mathrm{x}\right)}[\because \tan \left(2\mathrm{x}\right)=\frac{2\tan \left(\mathrm{x}\right)}{1-{\tan }^2\left(\mathrm{x}\right)}] \\ \Rightarrow \cot \left(4\mathrm{x}\right)=\frac{1-{\left(\frac{2\tan \left(\mathrm{x}\right)}{1-{\tan }^2\left(\mathrm{x}\right)}\right)}^2}{2·\frac{2\tan \left(\mathrm{x}\right)}{1-{\tan }^2\left(\mathrm{x}\right)}}[\because \tan \left(2\mathrm{x}\right)=\frac{2\tan \left(\mathrm{x}\right)}{1-{\tan }^2\left(\mathrm{x}\right)}] \\ \Rightarrow \cot \left(4\mathrm{x}\right)=\frac{{\left(1-{\tan }^2\left(\mathrm{x}\right)\right)}^2-{\left(2\tan \left(\mathrm{x}\right)\right)}^2}{4·\tan \left(\mathrm{x}\right)\left(1-{\tan }^2\left(\mathrm{x}\right)\right)} \\ \Rightarrow \cot \left(4\mathrm{x}\right)=\frac{1-2·{\tan }^2\left(\mathrm{x}\right)+{\tan }^4\left(\mathrm{x}\right)-4·{\tan }^2\left(\mathrm{x}\right)}{4·\tan \left(\mathrm{x}\right)-4·{\tan }^3\left(\mathrm{x}\right)}\left[\because {\left(\mathrm{a}-\mathrm{b}\right)}^2={\mathrm{a}}^2-2\mathrm{ab}+{\mathrm{b}}^2\right] \\ \Rightarrow \cot \left(4\mathrm{x}\right)=\frac{1-6·{\tan }^2\left(\mathrm{x}\right)+{\tan }^4\left(\mathrm{x}\right)}{4·\tan \left(\mathrm{x}\right)-4·{\tan }^3\left(\mathrm{x}\right)} \end{gathered}$$

Hence, $\cot \left(4\mathrm{x}\right)=\frac{1-6·{\tan }^2\left(\mathrm{x}\right)+{\tan }^4\left(\mathrm{x}\right)}{4·\tan \left(\mathrm{x}\right)-4·{\tan }^3\left(\mathrm{x}\right)}$.