Question

If $cosec\theta -\cot \theta =\frac{1}{3}$, then $\theta$ lies in

• A. $Q_1$
• B. $Q_2$
• C. $Q_3$
• D. $Q_4$

Answer 1

Answer: (A)

$Q_1$

$\text{cosec}\theta -\cot \theta =\frac{1}{3}$

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

$⟹\frac{2{\sin }^2\theta /2}{2\sin \theta /2\cos \theta /2}=\frac{1}{3}$

$⟹\tan \frac{\theta }{2}=\frac{1}{3}\cdots \left(1\right)$

$⟹\tan \theta =\frac{2\times \frac{1}{3}}{1-\frac{1}{9}}=\frac{3}{4}\cdots \left(2\right)$

$⟹\theta$ lies in $Q_1,Q_3$

and also from $\left(1\right),\frac{\theta }{2}$ lies in $Q_1,Q_3$

So $\theta$ lies in $Q_1$