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Question Number 167911 by MathsFan last updated on 29/Mar/22

I f a^{2} + b^{2} = 1

t h e n, \frac{1 + b + i a}{1 + b - i a} = ?

Answered by MJS_new last updated on 29/Mar/22

\frac{b + 1 + a i}{b + 1 - a i} = \frac{{(b + 1 + a i)}^{2}}{(b + 1 - a i) (b + 1 + a i)} =

= \frac{- a^{2} + b^{2} + 2 b + 1 + 2 a (b + 1) i}{a^{2} + b^{2} + 2 b + 1} =

[b^{2} = 1 - a^{2}]

= \frac{- 2 a^{2} + 2 b + 2 + 2 a (b + 1) i}{2 b + 2} =

= \frac{- a^{2} + b + 1}{b + 1} + a i =

[a^{2} = 1 - b^{2}]

= \frac{b^{2} + b}{b + 1} + a i =

= b + a i

Commented by MathsFan last updated on 29/Mar/22

t h a n k s