Question

If tan A + cot A = 4, then tan⁴ A + cot⁴ A is equal to
(a) 110
(b) 191
(c) 80
(d) 194

Answer 1

(d) 194

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \tan \mathrm{A}+\cot \mathrm{A}=4 \\ \mathrm{Squaring}\mathrm{both}\mathrm{the}\mathrm{sides}: \\ {\left(\tan \mathrm{A}+\cot \mathrm{A}\right)}^2=4^2 \\ \Rightarrow {\tan }^2\mathrm{A}+{\cot }^2\mathrm{A}+2\left(\tan \mathrm{A}\right)\left(\cot \mathrm{A}\right)=16 \\ \Rightarrow {\tan }^2\mathrm{A}+{\cot }^2\mathrm{A}+2=16 \\ \Rightarrow {\tan }^2\mathrm{A}+{\cot }^2\mathrm{A}=14 \\ \mathrm{Squaring}\mathrm{both}\mathrm{the}\mathrm{sides}\mathrm{again}: \\ {\left({\tan }^2\mathrm{A}+{\cot }^2\mathrm{A}\right)}^2={14}^2 \\ \Rightarrow {\tan }^4\mathrm{A}+{\cot }^4\mathrm{A}+2\left({\tan }^2\mathrm{A}\right)\left({\cot }^2\mathrm{A}\right)=196 \\ \Rightarrow {\tan }^4\mathrm{A}+{\cot }^4\mathrm{A}+2=196 \\ \Rightarrow {\tan }^4\mathrm{A}+{\cot }^4\mathrm{A}=194 \end{gathered}$$