<s>[INST] How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$? How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$? 
What else? Well.
Well can you use long division?
 [/INST] Use method add something and take something.
$$\frac{t}{t-1}= \frac{t-1+1}{t-1}$$
Then splits equation on 2 parts.
$$\frac{t-1}{t-1}+ \frac{1}{t-1}  $$ 
Then reduce fraction and you get:  $$1+\frac{1}{t-1}$$
If this is true
$$\frac{t}{t-1}=  1-\frac{1}{t-1}$$
Than you get:
$$\frac{t}{t-1}=  \frac{t-1-1}{t-1}  $$ 
Than this equation  multiply with $(t-1)$
$$t=t-1-1 \Rightarrow t=t-2 \Rightarrow 0t=-2$$  
So we take  $t=t-2$
$$\frac{t-2}{t-2-1}= \frac{t-2-1+1}{t-2-1}$$
$$\frac{t-2-1}{t-2-1}+ \frac{1}{t-2-1}$$
We get solution which is not the same as we previously calculated
$$1 + \frac{1}{t-2-1}=1 + \frac{1}{t-3}$$
So
$$\frac{t}{t-1} \neq 1 -\frac{1}{t-1}$$

$$\frac{t}{t+1} = \frac{t+1-1}{t+1} = \frac{t+1}{t+1} -\frac{1}{t+1}= 1 -\frac{1}{t+1} $$
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