if a² + b² =1 , then find the value of 1+b+ia/1+b-ia

the given expression is: $\frac{1+b+ia}{1+b-ia}$ 
multiplying by (b+ia) to the numerator and denominator:

$$\begin{gathered} \frac{1+b+ia}{1+b-ia}\times \frac{b+ia}{b+ia} \\ =\frac{(1+b+ia)\times (b+ia)}{b+b^2-iab+ia+iab+a^2} \\ =\frac{(1+b+ia)\times (b+ia)}{a^2+b^2+b+ia} \\ =\frac{(1+b+ia)\times (b+ia)}{1+b+ia} [given a^2+b^2=1] \end{gathered}$$
$=b+ia$

hope this helps you