Solve left(x-y/x+y+x+y/x-yright)left(x^2+y^2/2xy+1right)xy/x^2+y^2

$$\left( \frac{ x-y }{ x+y } + \frac{ x+y }{ x-y } \right) \left( \frac{ { x }^{ 2 } + { y }^{ 2 } }{ 2xy } +1 \right) \frac{ xy }{ { x }^{ 2 } + { y }^{ 2 } }$$

Evaluate

$$\frac{x+y}{x-y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x+y}{x-y} times \frac{x+y}{x+y}.

$$\left(\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}+\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}\right)\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

Since \frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-y\right)\left(x-y\right)+\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

Do the multiplications in \left(x-y\right)\left(x-y\right)+\left(x+y\right)\left(x+y\right).

$$\frac{x^{2}-xy-xy+y^{2}+x^{2}+xy+xy+y^{2}}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

Combine like terms in x^{2}-xy-xy+y^{2}+x^{2}+xy+xy+y^{2}.

$$\frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2xy}{2xy}.

$$\frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+\frac{2xy}{2xy}\right)\times \frac{xy}{x^{2}+y^{2}}$$

Since \frac{x^{2}+y^{2}}{2xy} and \frac{2xy}{2xy} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)}\times \frac{x^{2}+y^{2}+2xy}{2xy}\times \frac{xy}{x^{2}+y^{2}}$$

Multiply \frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)} times \frac{x^{2}+y^{2}+2xy}{2xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+y^{2}+2xy\right)}{\left(x+y\right)\left(x-y\right)\times 2xy}\times \frac{xy}{x^{2}+y^{2}}$$

Multiply \frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+y^{2}+2xy\right)}{\left(x+y\right)\left(x-y\right)\times 2xy} times \frac{xy}{x^{2}+y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+y^{2}+2xy\right)xy}{\left(x+y\right)\left(x-y\right)\times 2xy\left(x^{2}+y^{2}\right)}$$

Cancel out xy in both numerator and denominator.

$$\frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+2xy+y^{2}\right)}{2\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x+y\right)^{2}\left(x^{2}+y^{2}\right)}{2\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Cancel out 2\left(x+y\right)\left(x^{2}+y^{2}\right) in both numerator and denominator.

$$\frac{x+y}{x-y}$$

Expand

$$\frac{x+y}{x-y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x+y}{x-y} times \frac{x+y}{x+y}.

$$\left(\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}+\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}\right)\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

Since \frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-y\right)\left(x-y\right)+\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

Do the multiplications in \left(x-y\right)\left(x-y\right)+\left(x+y\right)\left(x+y\right).

$$\frac{x^{2}-xy-xy+y^{2}+x^{2}+xy+xy+y^{2}}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

Combine like terms in x^{2}-xy-xy+y^{2}+x^{2}+xy+xy+y^{2}.

$$\frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+1\right)\times \frac{xy}{x^{2}+y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2xy}{2xy}.

$$\frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)}\left(\frac{x^{2}+y^{2}}{2xy}+\frac{2xy}{2xy}\right)\times \frac{xy}{x^{2}+y^{2}}$$

Since \frac{x^{2}+y^{2}}{2xy} and \frac{2xy}{2xy} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)}\times \frac{x^{2}+y^{2}+2xy}{2xy}\times \frac{xy}{x^{2}+y^{2}}$$

Multiply \frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)} times \frac{x^{2}+y^{2}+2xy}{2xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+y^{2}+2xy\right)}{\left(x+y\right)\left(x-y\right)\times 2xy}\times \frac{xy}{x^{2}+y^{2}}$$

Multiply \frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+y^{2}+2xy\right)}{\left(x+y\right)\left(x-y\right)\times 2xy} times \frac{xy}{x^{2}+y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+y^{2}+2xy\right)xy}{\left(x+y\right)\left(x-y\right)\times 2xy\left(x^{2}+y^{2}\right)}$$

Cancel out xy in both numerator and denominator.

$$\frac{\left(2x^{2}+2y^{2}\right)\left(x^{2}+2xy+y^{2}\right)}{2\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x+y\right)^{2}\left(x^{2}+y^{2}\right)}{2\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Cancel out 2\left(x+y\right)\left(x^{2}+y^{2}\right) in both numerator and denominator.

$$\frac{x+y}{x-y}$$