If \( (a+i b)=\frac{c+i}{c-i} \), where \( c \) is real, prove that \( a^{2}+b^{2}=1 \) and \( \...

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If \( (a+i b)=\frac{c+i}{c-i} \), where \( c \) is real, prove that \( a^{2}+b^{2}=1 \) and \( \frac{b}{a}=\frac{2 c}{c^{2}-1} \).
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