Discussion Forum

1)

The positive integer value of n > 3 satisfying the equation

$\frac{1}{\sin\left(\frac{\pi}{n}\right)}=\frac{1}{\sin\left(\frac{2\pi}{n}\right)}+\frac{1}{\sin\left(\frac{3\pi}{n}\right)}$ is 

Answer:

Option C

Explanation:

 Given ,n>3 $\epsilon$ integer

 and $\frac{1}{\sin\left(\frac{\pi}{n}\right)}=\frac{1}{\sin\left(\frac{2\pi}{n}\right)}+\frac{1}{\sin\left(\frac{3\pi}{n}\right)}$

 $\Rightarrow$  $\frac{1}{\sin\frac{\pi}{n}}-\frac{1}{\sin\frac{3\pi}{n}}=\frac{1}{\sin\frac{2\pi}{n}}$

$\Rightarrow$   $\frac{\sin\frac{3\pi}{n}-\sin\frac{\pi}{n}}{\sin\frac{\pi}{n}.\sin\frac{3\pi}{n}}=\frac{1}{\sin\frac{2\pi}{n}}$

$\Rightarrow$  $2 \cos \left(\frac{2 \pi}{n}\right).\sin \frac{\pi}{n}=\frac{\sin\frac{\pi}{n}.\sin\frac{3\pi}{n}}{\sin\frac{2\pi}{n}}$

$\Rightarrow$  $2 \sin \frac{2 \pi}{n}.\cos \frac{2\pi}{n}= \sin \frac{3 \pi}{n}$

$\Rightarrow$  $\sin \frac{4 \pi}{3}=\sin  \frac{ 3 \pi}{n}$

$\Rightarrow$   $\frac{ 4 \pi}{n}= \pi-\frac{3 \pi}{n} \Rightarrow  \frac{7 \pi}{n} = \pi   \Rightarrow  n=7$