2012/007) find n when ( cosx + isinx )^n . ( sinx + icosx )^n = -1

( cosx + isinx )^n . ( sinx + icosx )^n
= (( cosx + isinx ) ( sinx + icosx ))^n
= ( cos x + i sin x) i ( - i sin x+ cos x ))^n
= (i ( cos x + i sin x)(cos x - i sin x))^n
= ( i ( cos ^2 x + sin ^2 x))^n
= i^n = - 1

as i^2 = 1 and i^4 = i

Hence n = 4m+2, for m∈Z