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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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