Divisibility of 2222^5555+5555^2222 by 7

For any prime p, square root of p is irrational.

Complex Numbers in Solving Trigonometrical problems

Show that $ \cos \frac{\pi}{11}+\cos \frac{3\pi}{11}+\cos \frac{5\pi}{11}+\cos 7\frac{\pi}{11}+\cos \frac{9\pi}{11}= \frac{1}{2}$ $$$$ I have used complex numbers to solve the problems. Standard trigonometric methods will be very much difficult to apply and will involve lots of calculation!!! Also it is important to link the different topics and see the beauty of mathematics! Important problem for students in 10+2 level $$$$ Also the idea used to solve the problem can be used to calculate the value of $ \sum_{i=1}^{\frac{n-1}{2}} \cos \frac{(2i-1) \pi}{n}$ where $n$ is an odd positive integer. $$$$ See the video below for the explanation. $$$$