Question

If $\sin \theta +\cos \theta =\sqrt{3}$, then prove that $\tan \theta +\cot \theta =1$

Answer 1

We have

$\sin \theta +\cos \theta =\sqrt{3}$

$\Rightarrow {\left(\sin \theta +\cos \theta \right)}^2=3$

$\Rightarrow {\sin }^2\theta +{\cos }^2\theta +2\sin \theta .\cos \theta =3$

$\Rightarrow 1+2\sin \theta .\cos \theta =3\therefore \left[{\sin }^2\theta +{\cos }^2\theta =1\right]$

$\Rightarrow 2\sin \theta .\cos \theta =2$

$\Rightarrow \sin \theta .\cos \theta =1={\sin }^2\theta +{\cos }^2\theta$

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

$\Rightarrow 1=\tan \theta +\cot \theta$

Hence,

$\tan \theta +\cot \theta =1$

Proved.