Solve x+y/x-y+x-y/x+y/frac{1{(x+y)^2}+1/(x-y)^2}

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

Evaluate

$$x^{2}-y^{2}$$

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}.

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

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.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+y\right)^{2} and \left(x-y\right)^{2} is \left(x+y\right)^{2}\left(x-y\right)^{2}. Multiply \frac{1}{\left(x+y\right)^{2}} times \frac{\left(x-y\right)^{2}}{\left(x-y\right)^{2}}. Multiply \frac{1}{\left(x-y\right)^{2}} times \frac{\left(x+y\right)^{2}}{\left(x+y\right)^{2}}.

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

Since \frac{\left(x-y\right)^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}} and \frac{\left(x+y\right)^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-y\right)^{2}+\left(x+y\right)^{2}.

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

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

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

Divide \frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)} by \frac{2x^{2}+2y^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}} by multiplying \frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{2x^{2}+2y^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}}.

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

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

$$\left(x+y\right)\left(x-y\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x^{2}-y^{2}$$

Expand

$$x^{2}-y^{2}$$

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}.

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

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.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+y\right)^{2} and \left(x-y\right)^{2} is \left(x+y\right)^{2}\left(x-y\right)^{2}. Multiply \frac{1}{\left(x+y\right)^{2}} times \frac{\left(x-y\right)^{2}}{\left(x-y\right)^{2}}. Multiply \frac{1}{\left(x-y\right)^{2}} times \frac{\left(x+y\right)^{2}}{\left(x+y\right)^{2}}.

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

Since \frac{\left(x-y\right)^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}} and \frac{\left(x+y\right)^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-y\right)^{2}+\left(x+y\right)^{2}.

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

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

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

Divide \frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)} by \frac{2x^{2}+2y^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}} by multiplying \frac{2x^{2}+2y^{2}}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{2x^{2}+2y^{2}}{\left(x+y\right)^{2}\left(x-y\right)^{2}}.

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

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

$$\left(x+y\right)\left(x-y\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x^{2}-y^{2}$$