Question

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

Answer 1

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

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

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

$1+2\sin \theta \cos \theta =3$

$2\sin \theta \cos \theta =2$

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

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

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

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