show that x power -1 + Y power minus one upon x power minus one plus x power minus 1 minus y power minus one upon x power minus 1 is equals to X square + Y square upon the xy
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Given : (x⁻¹ + y⁻¹)/x⁻¹ + (x⁻¹ - y⁻¹)/y⁻¹ = (x² + y²)/xy
To find : Prove the equality
Solution:
(x⁻¹ + y⁻¹)/x⁻¹ + (x⁻¹ - y⁻¹)/y⁻¹ = (x² + y²)/xy
LHS
= (x⁻¹ + y⁻¹)/x⁻¹ + (x⁻¹ - y⁻¹)/y⁻¹
= (1/x + 1/y)/(1/x) + (1/x - 1/y)/(1/y)
= x(y + x) /xy + y(y - x)/xy
= (1/xy) ( xy + x² + y² - yx)
= (1/xy) ( x² + y²)
= (x² + y²)/xy
= RHS
QED
Hence Proved
(x⁻¹ + y⁻¹)/x⁻¹ + (x⁻¹ - y⁻¹)/y⁻¹ = (x² + y²)/xy
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