Solve (1-(x)^(2))/(1+(x)^(2))\times((1)/((x-1)^(2))-(x)/(1-(x)^(2))) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$-\frac{1}{x-1}$

Short Solution Steps

Factor $1-x^{2}$.

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

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

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

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

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

Do the multiplications in $x+1-x\left(-1\right)\left(x-1\right)$.

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

Extract the negative sign in $-1-x$.

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

Cancel out $\left(x-1\right)\left(x+1\right)$ in both numerator and denominator.

$$\frac{-1}{x-1}$$

Expand

$-\frac{1}{x-1}$

Short Solution Steps

Factor $1-x^{2}$.

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

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

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

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

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

Do the multiplications in $x+1-x\left(-1\right)\left(x-1\right)$.

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

Extract the negative sign in $-1-x$.

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

Cancel out $\left(x-1\right)\left(x+1\right)$ in both numerator and denominator.

$$\frac{-1}{x-1}$$