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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

$$\frac{t^{4}-2t^{3}+2t^{2}+2t^{3}-4t^{2}+4t+t^{2}-2t+2+t^{4}+2t^{3}+2t^{2}-2t^{3}-4t^{2}-4t+t^{2}+2t+2}{\left(\left(t-1\right)^{2}+1\right)\left(\left(t+1\right)^{2}+1\right)}$$

Combine like terms in t^{4}-2t^{3}+2t^{2}+2t^{3}-4t^{2}+4t+t^{2}-2t+2+t^{4}+2t^{3}+2t^{2}-2t^{3}-4t^{2}-4t+t^{2}+2t+2.

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

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

$$\frac{2t^{4}-2t^{2}+4}{t^{4}+4}$$

Expand

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

Short Solution Steps

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

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

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

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

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

$$\frac{t^{4}-2t^{3}+2t^{2}+2t^{3}-4t^{2}+4t+t^{2}-2t+2+t^{4}+2t^{3}+2t^{2}-2t^{3}-4t^{2}-4t+t^{2}+2t+2}{\left(\left(t-1\right)^{2}+1\right)\left(\left(t+1\right)^{2}+1\right)}$$

Combine like terms in t^{4}-2t^{3}+2t^{2}+2t^{3}-4t^{2}+4t+t^{2}-2t+2+t^{4}+2t^{3}+2t^{2}-2t^{3}-4t^{2}-4t+t^{2}+2t+2.

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

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

$$\frac{2t^{4}-2t^{2}+4}{t^{4}+4}$$