Solve {left(frac{1-x^2/1+x^2right)}^2*2/1+x^2}{{left(1+1-x^2/1+{x^2}+2x/1+{x^2}right)}^2}

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

Evaluate

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

Short Solution Steps

To raise \frac{1-x^{2}}{1+x^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{\left(1-x^{2}\right)^{2}}{\left(1+x^{2}\right)^{2}} times \frac{2}{1+x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Since \frac{1+x^{2}}{1+x^{2}} and \frac{1-x^{2}}{1+x^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in 1+x^{2}+1-x^{2}.

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

Since \frac{2}{1+x^{2}} and \frac{2x}{1+x^{2}} have the same denominator, add them by adding their numerators.

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

To raise \frac{2+2x}{1+x^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Divide \frac{\left(1-x^{2}\right)^{2}\times 2}{\left(1+x^{2}\right)^{2}\left(1+x^{2}\right)} by \frac{\left(2+2x\right)^{2}}{\left(1+x^{2}\right)^{2}} by multiplying \frac{\left(1-x^{2}\right)^{2}\times 2}{\left(1+x^{2}\right)^{2}\left(1+x^{2}\right)} by the reciprocal of \frac{\left(2+2x\right)^{2}}{\left(1+x^{2}\right)^{2}}.

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

Cancel out \left(x^{2}+1\right)^{2} in both numerator and denominator.

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

Use the distributive property to multiply \left(2x+2\right)^{2} by x^{2}+1.

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

Factor the expressions that are not already factored.

$$\frac{2\left(-x^{2}+1\right)^{2}}{4\left(x+1\right)^{2}\left(x^{2}+1\right)}$$

Cancel out 2 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{4}-2x^{2}+1}{2x^{4}+4x^{3}+4x^{2}+4x+2}$$

Expand

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

Short Solution Steps

To raise \frac{1-x^{2}}{1+x^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{\left(1-x^{2}\right)^{2}}{\left(1+x^{2}\right)^{2}} times \frac{2}{1+x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Since \frac{1+x^{2}}{1+x^{2}} and \frac{1-x^{2}}{1+x^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in 1+x^{2}+1-x^{2}.

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

Since \frac{2}{1+x^{2}} and \frac{2x}{1+x^{2}} have the same denominator, add them by adding their numerators.

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

To raise \frac{2+2x}{1+x^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Divide \frac{\left(1-x^{2}\right)^{2}\times 2}{\left(1+x^{2}\right)^{2}\left(1+x^{2}\right)} by \frac{\left(2+2x\right)^{2}}{\left(1+x^{2}\right)^{2}} by multiplying \frac{\left(1-x^{2}\right)^{2}\times 2}{\left(1+x^{2}\right)^{2}\left(1+x^{2}\right)} by the reciprocal of \frac{\left(2+2x\right)^{2}}{\left(1+x^{2}\right)^{2}}.

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

Cancel out \left(x^{2}+1\right)^{2} in both numerator and denominator.

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

Use the distributive property to multiply \left(2x+2\right)^{2} by x^{2}+1.

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

Factor the expressions that are not already factored.

$$\frac{2\left(-x^{2}+1\right)^{2}}{4\left(x+1\right)^{2}\left(x^{2}+1\right)}$$

Cancel out 2 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{4}-2x^{2}+1}{2x^{4}+4x^{3}+4x^{2}+4x+2}$$