Question

$\mathrm{If}\cos \theta =\frac{7}{25},\mathrm{then}\mathrm{write}\mathrm{the}\mathrm{value}\mathrm{of}\left(\tan \theta +\cot \theta \right).\left[\mathrm{CBSE}2008\right]$

Answer 1

$$\begin{gathered} \mathrm{As}{\sin }^2\theta =1-{\cos }^2\theta \\ =1-{\left(\frac{7}{25}\right)}^2 \\ =1-\frac{49}{625} \\ =\frac{625-49}{625} \\ \Rightarrow {\sin }^2\theta =\frac{576}{625} \\ \Rightarrow \sin \theta =\sqrt{\frac{576}{625}} \\ \Rightarrow \sin \theta =\frac{24}{25} \end{gathered}$$

$$\begin{gathered} \mathrm{Now}, \\ \tan \theta +\cot \theta \\ =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta } \\ =\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta \sin \theta } \\ =\frac{1}{\left({\displaystyle \frac{7}{25}}\times {\displaystyle \frac{24}{25}}\right)} \\ =\frac{1}{\left({\displaystyle \frac{168}{625}}\right)} \\ =\frac{625}{168} \end{gathered}$$