algebric identity

If  ^x/_y + ^y/_x = -1, then find the value of x³ –y³

Answer:

(x - y)³ = x³ - 3x²y + 3xy² - y³

Or, (x - y)³ =  x³ – 3xy(x - y)  – y³

Or, x³– y³ = (x - y)³ + 3xy(x - y)

Or, x³– y³ = (x - y)[(x - y)²  + 3xy]       ----------(1)

Since, ^x/_y + ^y/_x = -1, multiply throughout by xy, we get

x² + y² = -xy    -----------(2)

Also, (x- y)² = x² + y² – 2xy = -xy – 2xy = -3xy  (by substituting (2)

Substituting the value of (x- y)²  in (1), we get

x³– y³ = (x - y)[ -3xy + 3xy]   = 0