Question

If $\text{cosec}A=2$, find the value of $\frac{1}{\tan A}+\frac{\sin A}{1+\cos A}$

Answer 1

Given, $\csc A=2$

We know,

$\sin A=\frac{1}{\csc A}=\frac{1}{2}$

And, $\sin 30=\frac{1}{2}$

$A=30$

$\tan 30=\frac{1}{\sqrt{3}}$ and $\cos 30=\frac{\sqrt{3}}{2}$

To find the value of $\frac{1}{\tan A}+\frac{\sin A}{1+\cos A}$

$=\frac{1}{\tan 30}+\frac{\sin 30}{1+\cos 30}$

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

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

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

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

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