Question

If $\sin \theta +\cos \theta =\surd 2$, evaluate $\tan \theta +\cot \theta$

Answer 1

Given:- $\sin \theta +\cos \theta =\sqrt{2}$

To find:- $\tan \theta +\cot \theta =?$

Sol:-

$\sin \theta +\cos \theta =\sqrt{2}$

Squaring bth side, we get

${\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta =2$

$2\sin \theta \cos \theta =2-1$

$\Rightarrow \sin \theta \cos \theta =\frac{1}{2}$

Therefore,

$\tan \theta +\cot \theta$

$=\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }$

$=\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }$

$=\frac{1}{\left(\frac{1}{2}\right)}=2$