Question

Roots of the equations are $(z+1{)}^5=(z-1{)}^5$ are

• A. $\pm i\tan (\frac{\pi }{5}),\pm i\tan (\frac{2\pi }{5})$
• B. $\pm i\cot (\frac{\pi }{5}),\pm i\cot (\frac{2\pi }{5})$
• C. $\pm i\cot (\frac{\pi }{5}),\pm i\tan (\frac{2\pi }{5})$
• D. None of these

Answer 1

The correct option is B $\pm i\cot (\frac{\pi }{5}),\pm i\cot (\frac{2\pi }{5})$

For $z\ne 1$, the given equation can be written as
${\left(\frac{z+1}{z-1}\right)}^5=1$
$\Rightarrow \frac{z+1}{z-1}=e^{2k\pi i/5}$
where $k=-2,-1,1,2$.
If we denote this value of $z$ by $z_k$, then
$z_k=\frac{e^{2k\pi i/5}+1}{e^{2k\pi i/5}-1}$
$=\frac{e^{k\pi i/5}+e^{-k\pi i/5}}{e^{k\pi i/5}-e^{-k\pi i/5}}$
$=-i\cot \left(\frac{k\pi }{5}\right),k=-2,-1,1,2$
Therefore, roots of the equation are
$\pm i\cot \left(\pi /5\right),\pm i\cot \left(2\pi /5\right)$.