Question

If $\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$, Find the value of $\theta$

Answer 1

$\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$

$\Rightarrow \frac{1-\tan \theta }{1+\tan \theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$

$\Rightarrow (1-\tan \theta )(1+\sqrt{3})=(1+\tan \theta )(1-\sqrt{3})$

$\Rightarrow (1+\sqrt{3})-\tan \theta (1+\sqrt{3})=(1-\sqrt{3})+\tan \theta (1-\sqrt{3})$

$\Rightarrow 1+\sqrt{3}-1+\sqrt{3}=\tan \theta (1-\sqrt{3}+1+\sqrt{3})$

$\Rightarrow \tan \theta =\frac{2\sqrt{3}}{2}$

$\Rightarrow \tan \theta =\sqrt{3}$

$\Rightarrow \theta =\frac{\pi }{3}$