Question

x^-1/x^-1 y^-1 + x^-1/ x^-1-y^-1
please provide me a short trick for this question..
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Answer 1

Step-by-step explanation:

Given : (x⁻¹ + y⁻¹)/x⁻¹ + (x⁻¹ - y⁻¹)/y⁻¹ = (x² + y²)/xy

To find : Prove the equality

Solution:

Correction in Question : other wise LHS = 2

(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