Solve 1/1+t+1/1-t-2/3t^2=

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

Evaluate

$$\frac{2}{3\left(1-t^{2}\right)}$$

Short Solution Steps

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

$$\frac{\frac{-t+1}{\left(t+1\right)\left(-t+1\right)}+\frac{t+1}{\left(t+1\right)\left(-t+1\right)}-2}{3t^{2}}$$

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

$$\frac{\frac{-t+1+t+1}{\left(t+1\right)\left(-t+1\right)}-2}{3t^{2}}$$

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

$$\frac{\frac{2}{\left(t+1\right)\left(-t+1\right)}-2}{3t^{2}}$$

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

$$\frac{\frac{2}{\left(t+1\right)\left(-t+1\right)}-\frac{2\left(t+1\right)\left(-t+1\right)}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

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

$$\frac{\frac{2-2\left(t+1\right)\left(-t+1\right)}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

Do the multiplications in 2-2\left(t+1\right)\left(-t+1\right).

$$\frac{\frac{2+2t^{2}-2t+2t-2}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

Combine like terms in 2+2t^{2}-2t+2t-2.

$$\frac{\frac{2t^{2}}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

Express \frac{\frac{2t^{2}}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}} as a single fraction.

$$\frac{2t^{2}}{\left(t+1\right)\left(-t+1\right)\times 3t^{2}}$$

Cancel out t^{2} in both numerator and denominator.

$$\frac{2}{3\left(t+1\right)\left(-t+1\right)}$$

Use the distributive property to multiply 3 by t+1.

$$\frac{2}{\left(3t+3\right)\left(-t+1\right)}$$

Use the distributive property to multiply 3t+3 by -t+1 and combine like terms.

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

Expand

$$\frac{2}{3\left(1-t^{2}\right)}$$

Short Solution Steps

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

$$\frac{\frac{-t+1}{\left(t+1\right)\left(-t+1\right)}+\frac{t+1}{\left(t+1\right)\left(-t+1\right)}-2}{3t^{2}}$$

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

$$\frac{\frac{-t+1+t+1}{\left(t+1\right)\left(-t+1\right)}-2}{3t^{2}}$$

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

$$\frac{\frac{2}{\left(t+1\right)\left(-t+1\right)}-2}{3t^{2}}$$

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

$$\frac{\frac{2}{\left(t+1\right)\left(-t+1\right)}-\frac{2\left(t+1\right)\left(-t+1\right)}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

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

$$\frac{\frac{2-2\left(t+1\right)\left(-t+1\right)}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

Do the multiplications in 2-2\left(t+1\right)\left(-t+1\right).

$$\frac{\frac{2+2t^{2}-2t+2t-2}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

Combine like terms in 2+2t^{2}-2t+2t-2.

$$\frac{\frac{2t^{2}}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}}$$

Express \frac{\frac{2t^{2}}{\left(t+1\right)\left(-t+1\right)}}{3t^{2}} as a single fraction.

$$\frac{2t^{2}}{\left(t+1\right)\left(-t+1\right)\times 3t^{2}}$$

Cancel out t^{2} in both numerator and denominator.

$$\frac{2}{3\left(t+1\right)\left(-t+1\right)}$$

Use the distributive property to multiply 3 by t+1.

$$\frac{2}{\left(3t+3\right)\left(-t+1\right)}$$

Use the distributive property to multiply 3t+3 by -t+1 and combine like terms.

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