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in this video we are going to find real numbers a b c d and e between -2 and 2 such that the sum equals 0 the sum of cubes also equals 0 but the sum of powers of 5 equals to 10. before i move on dont forget to give a like subscribe to my channel and turn on postifications there is an unusual trick that can help solve this problem and that is trigonometry precisely we can let each variable to be equal to two times cosine of some angle lets say a equals two times cos big a and im going to do similar things on the other unknowns b equals to correspect b c equals 2 cos c etc now to proceed im going to consider [Music] cos of 5a when the angle is multiplied by 5 and by some algebra which im going to skip in this video is that its actually equal to written in terms of cosine 16 times cos a all to power 5 minus 20 times cos a all cubed and then plus 5 cos a then the trick is that we can notice if i add up these expressions cos five a minus one cos five b minus one cos five c minus one