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Which of the following is equivalent to [tex] \frac{i}{1+i} - \frac{1}{2-i} [/tex]?

A. [tex] \frac{3i+1}{10} [/tex]
B. [tex] \frac{3+i}{i} [/tex]
C. [tex] \frac{4i}{10} [/tex]
D. [tex] \frac{i-2}{3+i} [/tex]


Sagot :

To determine which of the given expressions is equivalent to \(\frac{i}{1+i} - \frac{1}{2-i}\), we perform the following steps to simplify the expression:

1. Simplify \(\frac{i}{1+i}\):

Rationalize the denominator:
[tex]\[ \frac{i}{1+i} \cdot \frac{1-i}{1-i} = \frac{i(1-i)}{(1+i)(1-i)} = \frac{i - i^2}{1 - i^2} \][/tex]

Simplify \(i^2 = -1\):
[tex]\[ \frac{i - (-1)}{1 - (-1)} = \frac{i + 1}{2} \][/tex]

So,
[tex]\[ \frac{i}{1+i} = \frac{1 + i}{2} \][/tex]

2. Simplify \(\frac{1}{2-i}\):

Rationalize the denominator:
[tex]\[ \frac{1}{2-i} \cdot \frac{2+i}{2+i} = \frac{1(2+i)}{(2-i)(2+i)} = \frac{2+i}{4 - (-1)} \][/tex]

Simplify \(i^2 = -1\):
[tex]\[ \frac{2+i}{5} \][/tex]

So,
[tex]\[ \frac{1}{2-i} = \frac{2+i}{5} \][/tex]

3. Subtract the two simplified fractions:
[tex]\[ \frac{1+i}{2} - \frac{2+i}{5} \][/tex]

Find a common denominator:
[tex]\[ \text{Common denominator: } 10 \][/tex]
Convert each fraction:
[tex]\[ \frac{1+i}{2} = \frac{5(1+i)}{10} = \frac{5 + 5i}{10} \][/tex]
[tex]\[ \frac{2+i}{5} = \frac{2(2+i)}{10} = \frac{4 + 2i}{10} \][/tex]

Subtract the fractions:
[tex]\[ \frac{5 + 5i}{10} - \frac{4 + 2i}{10} = \frac{(5 + 5i) - (4 + 2i)}{10} = \frac{5 + 5i - 4 - 2i}{10} = \frac{1 + 3i}{10} \][/tex]

So,
[tex]\[ \frac{i}{1+i} - \frac{1}{2-i} = \frac{1 + 3i}{10} \][/tex]

Thus, the expression equivalent to \(\frac{i}{1+i} - \frac{1}{2-i}\) is:

[tex]\[ \boxed{\frac{3i+1}{10}} \][/tex]
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