Summation of cosine value from 0degrees to 89degrees

Question

I am unable to solve the summation of cosine value from 0 to 89 degrees. Please help me. Thank you.

Answer 1

Easier using complex numbers.

$$\sum_{k=0}^{89}e^{ik\pi/180}=\frac{e^{i\pi/2}-1}{e^{i\pi/180}-1}=\frac{(i-1)(e^{-i\pi/180}-1)}{|e^{i\pi/180}-1|^2}.$$

Then the real part is

$$\frac{1-\cos\frac{\pi}{180}+\sin\frac{\pi}{180}}{(\cos\frac\pi{180}-1)^2+\sin^2\frac\pi{180}}.$$

The sum of the sines comes as a byproduct.