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On observing , (x²+1/x²)(x³+1/x³) = x^5+1/x^5 +x+1/x
x²+1/x² = (x+1/x)² - 2 = 1-2 = -1
x³+1/x³ = (x+1/x)(x²+1/x² -1) = (1)(-1-1) = -2
So , x^5+1/x^5 +x+1/x = (-1)(-2) = 2
x^5 + 1/x^5 = 2-1 = 1
x²+1/x² = (x+1/x)² - 2 = 1-2 = -1
x³+1/x³ = (x+1/x)(x²+1/x² -1) = (1)(-1-1) = -2
So , x^5+1/x^5 +x+1/x = (-1)(-2) = 2
x^5 + 1/x^5 = 2-1 = 1
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