Question

Express the following in `a + ib` form:
(i) `((cos theta+ isin theta)/(sin theta + icos theta))^(4)`
(ii)` ((cos 2theta -isin2theta)^(4)(cos4theta +isin4theta)^(-5))/((cos 3theta +isin3theta)^(-2) (cos 3theta -isin 3theta)^(-9))`
(iii) ` ((sinpi//8 +icos pi//8)^(8))/((sinpi//8-icospi//8)^(8))`

Answer 1

(i) `((costheta+isintheta)/(sintheta+icostheta))^(4)=((costheta+isintheta)^(4))/(i^(4)(costheta-isintheta)^(4))`
`=((costheta+isintheta)^(4))/((costheta+isintheta)^(-4))`
`=(cos8theta+isintheta)^(8)`
`=cos8theta+isin8theta`
(ii) `((cos2theta-isin2theta)^(4)(cos4theta+isin4theta)^(-5))/((cos3theta+isin3theta)^(-2)(cos3theta-isin3theta)^(-9))`
`=([(costheta+isintheta)^(-2)]^(4)[(costheta+isintheta)^(4)]^(-5))/([(costheta+isintheta)^(3)]^(-2)[(costheta+isintheta)^(-3)]^(-9))`
`=((costheta+isintheta)^(-8)(costheta+isintheta)^(-20))/((costheta+isintheta)^(-6)(costheta+isintheta)^(27))`
`=(costheta+isintheta)^(-8-20+6-27)`
`=(costheta+isintheta)^(-49)`
`=cos49theta-isin49theta`
(iii) `((sinpi//8+icospi//8)^(8))/((sinpi//8-icospi//8)^(8))`
`(i^(8)(cospi//8-isinpi//8)^(8))/((-i)^(8)(cospi//8+isinpi//8)^(8))=(cospi-isinpi)/(cospi+isinpi)=1`