Evaluate

We have, $\left(\frac{\mathrm{i}^{180}+\mathrm{i}^{178}+\mathrm{i}^{176}+\mathrm{i}^{174}+\mathrm{i}^{172}}{\mathrm{i}^{170}+\mathrm{i}^{168}+\mathrm{i}^{166}+\mathrm{i}^{164}+\mathrm{i}^{162}}\right)$

$=\left(\frac{\mathrm{i}^{180}+\mathrm{i}^{178}+\mathrm{i}^{176}+\mathrm{i}^{174}+\mathrm{i}^{172}}{\mathrm{i}^{170}+\mathrm{i}^{168}+\mathrm{i}^{166}+\mathrm{i}^{164}+\mathrm{i}^{162}}\right)$

$=\left(\frac{\left(i^{4}\right)^{45}+\left(i^{4}\right)^{44} \cdot i^{2}+\left(i^{4}\right)^{44}+\left(i^{4}\right)^{43} \cdot i^{2}+\left(i^{4}\right)^{43}}{\left(i^{4}\right)^{42} \cdot i^{2}+\left(i^{4}\right)^{42}+\left(i^{4}\right)^{41} \cdot i^{2}+\left(i^{4}\right)^{41}+\left(i^{4}\right)^{40} \cdot i^{2}}\right)$

$=\left(\frac{(1)^{45}+(1)^{44} \cdot i^{2}+(1)^{44}+(1)^{43} \cdot i^{2}+(1)^{43}}{(1)^{42} \cdot i^{2}+(1)^{42}+(1)^{41} \cdot i^{2}+(1)^{41}+(1)^{40} \cdot i^{2}}\right)$

$=\left(\frac{1+i^{2}+1+. i^{2}+1}{i^{2}+1+i^{2}+1+i^{2}}\right)$

$=\left(\frac{1-1+1-1+1}{-1+1-1+1-1}\right)$

$=\left(\frac{1}{-1}\right)$

$=-1$