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

(i) `(5+4i)/(4+5i) =(5+4i)/(4+5i)xx(4-5i)/(4-5i)`
` =((20 +20) +i(16-25))/(16-25i) =(40-9i)/(41) =(40)/(41) -(9)/(41)i`
(ii) `((1+i)^(2))/(3-i) =(1+2i+i^(2))/(3-i)`
`=(2i)/(3-i) =(2i)/(3-i) xx(3+i)/(3+i) = (6i+2i^(2))/(9-i)`
`=(-2+6i)/(10) = -(1)/(5) + (3)/(5) i`
(iii) `(1)/(1- cos thets + 2i sin theta) `
`= (1)/(1-cos theta + 2i sin theta) xx(1- cos theta - 2i sin theta)/(1- cos theta - 2i sin theta)`
`=(1- cos theta - 2i sin theta)/((1- cos theta)^(2) + 4 sin theta)`
`=
(1-cos theta- 2i sin theta)/(1-2 cos theta + cos^(2) theta + 4 sin^(2) theta)`
`= (1- cos theta - 2 i sin theta)/( 2-2 cos theta + 3 sin^(2) theta)`
`= ((1-cos theta)/( 2-2 sin theta + 3 sin^(2) theta))+i((-2 sin theta)/(2-2 cos theta + 3 sin^(2)theta))`