Express each of the following in the standerd from ` a + ib`
(i) `(5+4i)/(4+ 5i)` (ii) ` ((1 +i)^(2))/(3-i)` (iii)` (1)/(1-cos theta + 2i sintheta)`

Explanation

Let's solve each part of the question step by step to express them in the standard form \( a + ib \).

### Part (i): \(\frac{5 + 4i}{4 + 5i}\)

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator.

\[
\frac{5 + 4i}{4 + 5i} \cdot \frac{4 - 5i}{4 - 5i} = \frac{(5 + 4i)(4 - 5i)}{(4 + 5i)(4 - 5i)}
...