Question

If tan x + cot x = 4, then tan⁴x + cot⁴x = ___________.

Answer 1

Given tan x + cot x = 4
$$\begin{gathered} \mathrm{Since}{\left({\tan }^{2}x+{\cot }^{2}x\right)}^2={\tan }^{4}x+{\cot }^{4}x+2{\cot }^{2}x{\tan }^{2}x \\ \mathrm{i}.\mathrm{e}{\left({\tan }^{2}x+{\cot }^{2}x\right)}^2={\tan }^{4}x+{\cot }^{4}x+2 \\ \mathrm{i}.\mathrm{e}{\tan }^{4}x+{\cot }^{4}x={\left({\tan }^{2}x+{\cot }^{2}x\right)}^2-2\left(1\right) \\ \mathrm{also},{\left(\tan x+\cot x\right)}^2={\left(4\right)}^2 \\ \mathrm{i}.\mathrm{e}{\tan }^{2}x+{\cot }^{2}x+2\cot x\tan x=16 \end{gathered}$$
$$\begin{gathered} {\tan }^{2}x+{\cot }^{2}x+2=16 \\ {\tan }^{2}x+{\cot }^{2}x=14 \\ \therefore \mathrm{equation}\left(1\right),\mathrm{reduces}\mathrm{to} \\ {\tan }^{4}x+{\cot }^{4}x={\left(14\right)}^2-2 \\ \mathrm{i}.\mathrm{e}{\tan }^{4}x+{\cot }^{4}x=194 \end{gathered}$$