Question

If tan $\theta$ + cot $\theta$ = 2, find the value of tan² $\theta$ + cot² $\theta$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{tan\theta }+\mathrm{cot\theta }=2 \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{have}: \\ {\left(\mathrm{tan\theta }+\mathrm{cot\theta }\right)}^2=2^2 \\ \Rightarrow \tan {}^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2\tan \theta .\cot \theta =4\left[\because (\mathrm{a}+\mathrm{b}{)}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}\right] \\ \Rightarrow \tan {}^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2\times 1=4(\because \tan \mathrm{\theta }.\cot \mathrm{\theta }=1) \\ \Rightarrow \tan {}^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2=4 \\ \Rightarrow \tan {}^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }=2 \end{gathered}$$