Question

If $cosecA=2$, then find the value of $\cot A+\frac{\sin A}{1+\cos A}$

Answer 1

$cosecA=2$ or $\sin A=\frac{1}{2}$

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

$\therefore \cos A=\frac{\sqrt{3}}{2}$

Now,

$\cot A+\frac{\sin A}{1+\cos A}$

$=\cot {30}^{\circ }+\frac{\sin {30}^{\circ }}{1+\cos {30}^{\circ }}$

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

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

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

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