Solve 1/(1+t)^2+1/(1-t^2)

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

Evaluate

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

Short Solution Steps

Factor 1-t^{2}.

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

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

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

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

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

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

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

Combine like terms in t-1-t-1.

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

Expand \left(t-1\right)\left(t+1\right)^{2}.

$$\frac{-2}{t^{3}+t^{2}-t-1}$$

Expand

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

Short Solution Steps

Factor 1-t^{2}.

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

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

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

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

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

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

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

Combine like terms in t-1-t-1.

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

Expand \left(t-1\right)\left(t+1\right)^{2}.

$$\frac{-2}{t^{3}+t^{2}-t-1}$$