Roots of the equation are $(z+1)^5 = (z - 1)^5$ are
- $\pm i\tan { \left( \displaystyle \frac { \pi }{ 5 } \right) ,\pm{ i\tan { \left( \displaystyle \frac { 2 }{ 5 } \right) } } } $
- $\pm i{ \cot { \left(\displaystyle \frac { \pi }{ 5 } \right) ,\pm {i \cot { \left(\displaystyle \frac { 2\pi }{ 5 } \right) } } } }$
- $\pm i{ \cot { \left(\displaystyle \frac { \pi }{ 5 } \right) ,\pm {i \tan { \left( \displaystyle \frac { 2\pi }{ 5 } \right) } } } }$
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none of these
Set (z + 1)/(z - 1) equal to the non-real fifth roots of unity. Using the corresponding angle values gives roots ±i cot(pi/5) and ±i cot(2pi/5).
We divide both sides of the equation by (z minus 1) raised to the power of 5 to get ((z plus 1) divided by (z minus 1)) raised to the power of 5 equals 1. By De Moivre's theorem, this ratio must equal the fifth roots of unity, which are the complex exponentials e raised to the power of (2 k pi i divided by 5) for k from 1 to 4. Solving the equation (z plus 1) divided by (z minus 1) equals e raised to the power of (2 k pi i divided by 5) for z yields z equals (e raised to the power of (2 k pi i divided by 5) plus 1) divided by (e raised to the power of (2 k pi i divided by 5) minus 1). Converting these exponentials to trigonometric form using Euler's formula simplifies the expression to i times the cotangent of (k pi divided by 5), confirming the roots are plus or minus i times the cotangent of (pi divided by 5) and plus or minus i times the cotangent of (2 pi divided by 5).