Question

If $\sin \theta =\frac{4}{5}$ and $\theta \in [0,{90}^{\circ }]$, then the value of $\tan \theta +\cot \theta$ is

• A. $\frac{25}{12}$
• B. $\pm \frac{25}{12}$
• C. $\frac{23}{12}$
• D. $\pm \frac{23}{12}$

Answer 1

The correct option is A $\frac{25}{12}$

$\sin \theta =\frac{4}{5}\Rightarrow {\cos }^2\theta =1-{\sin }^2\theta \Rightarrow {\cos }^2\theta =1-\frac{16}{25}\Rightarrow \cos \theta =\pm \sqrt{\frac{9}{25}}\Rightarrow \cos \theta =\frac{3}{5}(\because \theta \in [0,{90}^{\circ }\left]\right)$

Now,
$\tan \theta +\cot \theta =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }=\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }=\frac{1}{\sin \theta \cos \theta }=\frac{25}{12}$