( + i + i )^4 equals - Vidyadip

MCQ

${\left( {\frac{{\cos \theta + i\sin \theta }}{{\sin \theta + i\cos \theta }}} \right)^4}$equals

A
$\sin 8\theta - i\cos 8\theta $
B
$\cos 8\theta - i\sin 8\theta $
C
$\sin 8\theta + i\cos 8\theta $
✓
$\cos 8\theta + i\sin 8\theta $

✓

Answer

Correct option: D.

$\cos 8\theta + i\sin 8\theta $

d
(d) ${\left( {\frac{{\cos \,\theta + i\sin \theta }}{{\sin \,\theta + i\cos \,\,\theta }}} \right)^4} = \frac{{{{(\cos \,\,\theta + i\sin \,\theta )}^4}}}{{{i^4}\,{{(\cos \,\,\theta - i\sin \theta )}^4}}}$$ = \frac{{\cos \,\,4\theta + i\sin \,\,4\theta }}{{\cos \,\,4\theta - i\sin \,\,4\theta }}$
$ = \frac{{(\cos 4\theta + i\sin 4\theta )(\cos 4\theta + i\sin 4\theta )}}{{(\cos 4\theta - i\sin 4\theta )(\cos 4\theta + i\sin 4\theta )}}$
$\frac{{{{(\cos 4\theta + i\sin 4\theta )}^2}}}{{{{\cos }^2}4\theta + {{\sin }^2}4\theta }} = \cos 8\theta + i\sin 8\theta $

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