The conjugate of a complex number is $$\frac{1}{i-1}$$. Then the complex number is

Let the required complex number be $$z$$. According to the statement
$$\overline{z}= \frac{1}{\,i-1\,}$$

Property used: For any non-zero complex number $$w$$, $$\displaystyle \overline{\frac{1}{w}}=\frac{1}{\,\overline{w}\,}$$ because conjugation preserves division and multiplication.

Applying the property, we get
$$z=\overline{\frac{1}{\,i-1\,}}=\frac{1}{\,\overline{i-1}\,}$$

Now $$i-1=-1+i$$ has real part $$-1$$ and imaginary part $$1$$, so its conjugate is
$$\overline{i-1}=-1-i=-(1+i)$$

Therefore
$$z=\frac{1}{-1-i}=\frac{-1}{1+i}=\frac{-1}{\,i+1\,}$$

The expression $$\displaystyle \frac{-1}{i+1}$$ exactly matches Option C.

Hence the complex number is $$\displaystyle \frac{-1}{i+1}$$.

Option C which is: $$\frac{-1}{i+1}$$