Question

If (tan θ + cot θ) = 5 then (tan² θ + cot² θ) = ?
(a) 27
(b) 25
(c) 24
(d) 23

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}\tan \theta +\cot \theta =5 \\ \tan \theta +\cot \theta =5 \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow {\left(\tan \theta +\cot \theta \right)}^2=5^2 \\ \Rightarrow {\tan }^2\theta +{\cot }^2\theta +2\left(\cot \theta \right)\left(\tan \theta \right)=25 \\ \Rightarrow {\tan }^2\theta +{\cot }^2\theta +2\left(\frac{1}{\tan \theta }\right)\left(\tan \theta \right)=25\left(\because \cot \theta =\frac{1}{\tan \theta }\right) \\ \Rightarrow {\tan }^2\theta +{\cot }^2\theta +2=25 \\ \Rightarrow {\tan }^2\theta +{\cot }^2\theta =23 \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{d}\right). \end{gathered}$$