Express the following in `a + ib` form: (a) `((cos alpha + i sin alpha)^(4))/((sin beta + i cos beta)^(5))` (b) `((1+ cos phi + i sin phi)/(1 + cos phi - isin phi))^(n)` (c) `((cos alpha + i sin alpha)(cos beta + i sin beta))/((cos gamma + i sin gamma)(cos delta + i sin delta))`

Answer

(a) ` sin(4 alpha + 5beta) - i cos ( 4alpha + 5 beta)`
(b) ` cos nphi+ isin nphi `
(c) `cos (alpha + beta -gamma -delta) + i sin (alpha + beta- gamma - delta)`

Explanation

`((cos alpha+sin alpha)^(4))/((sin beta +icos beta)^(5)) = (cos4alpha + isin 4alpha)/(i^(5)(cos beta -isin beta)^(5))`
`=-i(cos 4alpha + isin 4alpha)(cos beta -isin beta)^(5)`
` =- i[cos 4alpha + isin 4alpha][cos 5 beta + isin 5 beta]`
`= -i[cos(4alpha + 5beta)+isin(4alpha + 5beta)]`
`= sin (4alpha + 5beta)-icos (4alpha + 5beta)`
(b) `((1+cos phi + isin phi)/(1+cos phi -isin phi))^(n)`
`=[(2cos^(2) (phi//2) + isin (phi//2) cos(pi//2))/(2cos^(2)(phi//2) -2isin(phi//2) cos(phi//2))]^(n)`
`= [(cos(phi//2) +isin (phi//2) )/(cos(phi//2) -isin (phi//2))]^(n)`
`=[((cos phi+ sin phi)^(1//2))/((cos phi +isin phi)^(-1//2))]^(n)`
`=[cos phi + isin phi]^(n)`
`= cos n phi + isinnphi`