Question

If $\text{cosec}\theta \text{=}\text{2}$ then $\left(\text{cot}\theta \text{+}\frac{\text{sin}\theta }{\text{1 + cos}\theta }\right)$ =

Answer 1

Solution:

If $\cos ec\theta =2$

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

Reciprocal on both the sides, we get,

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

$\therefore \theta ={30}^{\circ }$

$\Rightarrow \frac{\sqrt{3}}{2}=\cos {30}^{\circ }$ or $\cos \theta$

$\Rightarrow \frac{\sqrt{3}}{1}=\cot {30}^{\circ }$ or $\cot \theta$

$\Rightarrow \cot \theta +\frac{\sin \theta }{1+\cos \theta }=\sqrt{3}+\frac{\frac{1}{2}}{1+\frac{\sqrt{3}}{2}}$

$=\sqrt{3}+\frac{1}{2}\times \frac{2}{2+\sqrt{3}}$

$=\sqrt{3}+\frac{1}{2+\sqrt{3}}$

$=\sqrt{3}+\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}$

$=\sqrt{3}+\frac{2-\sqrt{3}}{4-3}$

$=\sqrt{3}+2-\sqrt{3}$

$=2$