How do you simplify (x-y)/(x+y) + (x+y)/(x-y)

I'm supposed to simplify it by finding the least common denominator but I can't figure it out. Can you show me the steps to solve this please?

x-y.........x+y
—— +..———
x+y........x-y


Least common denominator
is (x+y)(x-y)

so above expression equals


(x-y)(x-y)..........(x+y)(x+y)
—————. +..—————
(x+y)(x-y)........(x-y)(x+y)


equals

(x-y)²................(x+y)²
—————. +..—————
(x+y)(x-y)........(x-y)(x+y)

equals

x²-2xy+y²............x²+2xy+y²
—————. +..—————
x²-y²......................x²-y²


equals

2x²+2y²
—————.
..x²-y²

You need a common denominator which can only be (x+y)(x-y) now look at what you have to do to each part of the expression to get that denominator. Multiply the left part by (x-y)/(x-y) which is actually just 1. Multiply the right part by (x+y)/(x+y) (Also 1)

(x-y)(x-y)/(x+y)(x-y) + (x+y)(x+y)/(x+y)(x-y) (multiply out the numerators)

[x^2 - 4y +y^2 + x^2 + 4y +y^2] /(x-y)(x+y) (collect numerator terms)

[2x^2 + 2y^2]/[(x +y)(x-y)] this reduces to

2(x^2 + y^2)/(x-y)(x+y) and multiplying out the denominator gives:

2(x^2 + y^2) / (x^2 - y^2)

lcd = (x + y)*(x - y)

Divide the lcd between each denominator and the remainder multiplied by each numerator; after, add them and divide between the product of the lcd.

((x - y)^2 + (x + y)^2)/((x + y)*(x - y)) = (x^2 - 2*x*y + y^2 + x^2 + 2*x*y + y^2)/((x + y)*(x - y)) =

(2*x^2 + 2*y^2)/(x^2 - y^2) = 2*(x^2 + y^2)/x^2 - y^2)

there is no denominator in this sum
can u verify the sum once again?

to solve

there is a formula a^2-b^2= (a+b)(a-b)
applying we get

(x^2-y^2)+(x^2-y^2)=2(x^2-y^2)

2(x²+y²) /x²-y²