Question

If $\mathrm{cosec}\left(\mathrm{A}\right)=2$ then obtain the value of $\frac{\cot \left(A\right)+\sin \left(A\right)}{1+\cos \left(A\right)}$

Answer 1

Find the required value:

Given $\mathrm{cosec}\left(\mathrm{A}\right)=2$

Now,

$$\begin{gathered} \frac{\cot \left(A\right)+\sin \left(A\right)}{1+\cos \left(A\right)} \\ =\frac{{\displaystyle \frac{\cos \left(A\right)}{\sin \left(A\right)}}+\sin \left(A\right)}{1+\cos \left(A\right)} \\ =\frac{\cos \left(A\right)+{\sin }^2\left(A\right)}{\left(1+\cos \left(A\right)\right)\sin \left(A\right)}......................\left(1\right) \end{gathered}$$

$\begin{array}{rcl}\mathrm{cosec}\left(\mathrm{A}\right)& =& 2\\ & \Rightarrow & \sin \left(A\right)=\frac{1}{2}\left[\because \sin \left(A\right)=\frac{1}{\mathrm{cosec}\left(\mathrm{A}\right)}\right]\\ \cos \left(A\right)& =& \sqrt{1-{\sin }^2\left(A\right)}\\ & =& \sqrt{1-\frac{1}{4}}\\ & =& \frac{\sqrt{3}}{2}\end{array}$

Putting the value of $\cos \left(A\right)$ and $\sin \left(A\right)$ in equation $\left(1\right)$ we get

$$\begin{gathered} \frac{\cos \left(A\right)+{\sin }^2\left(A\right)}{\left(1+\cos \left(A\right)\right)\sin \left(A\right)} \\ =\frac{{\displaystyle \frac{\sqrt{3}}{2}}+{\left({\displaystyle \frac{1}{2}}\right)}^2}{\left(1+\frac{\sqrt{3}}{2}\right)\frac{1}{2}} \\ =\frac{{\displaystyle \frac{2\sqrt{3}+1}{4}}}{{\displaystyle \frac{2+\sqrt{3}}{4}}} \\ =\frac{2\sqrt{3}+1}{2+\sqrt{3}} \\ =\frac{\left(2\sqrt{3}+1\right)\left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\left[\mathrm{Rationalizing}\right] \\ =\frac{4\sqrt{3}-6+2-\sqrt{3}}{4-3} \\ =3\sqrt{3}-4 \end{gathered}$$

Hence, the value of $\frac{\cot \left(A\right)+\sin \left(A\right)}{1+\cos \left(A\right)}$ is $3\sqrt{3}-4$.