Solve 1/1+4x^{2/(1-x^{2)^2}}*1+x^2/(1-x^2)^2

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

Evaluate

$$\frac{1}{x^{2}+1}$$

Short Solution Steps

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

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

Since \frac{\left(1-x^{2}\right)^{2}}{\left(1-x^{2}\right)^{2}} and \frac{4x^{2}}{\left(1-x^{2}\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(1-x^{2}\right)^{2}+4x^{2}.

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

Combine like terms in 1-2x^{2}+x^{4}+4x^{2}.

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

Divide 1 by \frac{1+2x^{2}+x^{4}}{\left(1-x^{2}\right)^{2}} by multiplying 1 by the reciprocal of \frac{1+2x^{2}+x^{4}}{\left(1-x^{2}\right)^{2}}.

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

Multiply \frac{\left(1-x^{2}\right)^{2}}{1+2x^{2}+x^{4}} times \frac{1+x^{2}}{\left(1-x^{2}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{1}{x^{2}+1}$$

Expand

$$\frac{1}{x^{2}+1}$$

Short Solution Steps

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

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

Since \frac{\left(1-x^{2}\right)^{2}}{\left(1-x^{2}\right)^{2}} and \frac{4x^{2}}{\left(1-x^{2}\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(1-x^{2}\right)^{2}+4x^{2}.

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

Combine like terms in 1-2x^{2}+x^{4}+4x^{2}.

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

Divide 1 by \frac{1+2x^{2}+x^{4}}{\left(1-x^{2}\right)^{2}} by multiplying 1 by the reciprocal of \frac{1+2x^{2}+x^{4}}{\left(1-x^{2}\right)^{2}}.

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

Multiply \frac{\left(1-x^{2}\right)^{2}}{1+2x^{2}+x^{4}} times \frac{1+x^{2}}{\left(1-x^{2}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{1}{x^{2}+1}$$