\(a)\,1 - i\tan {\pi \over 5} = 1 - i{{\sin {\pi \over 5}} \over {\cos {\pi \over 5}}} = {1 \over {\cos {\pi \over 5}}}\left( {\cos {\pi \over 5} - i\sin {\pi \over 5}} \right) = {1 \over {\cos {\pi \over 5}}}\left[ {\cos \left( { - {\pi \over 5}} \right) + i\sin \left( { - {\pi \over 5}} \right)} \right]\)
\(b)\,\tan {{5\pi } \over 8} + i = {{ - 1} \over {\cos {{5\pi } \over 8}}}\left( { - \sin {{5\pi } \over 8} - i\cos {{5\pi } \over 8}} \right)\)(để ý rằng \(\cos {{5\pi } \over 8} < 0\))
\( = {1 \over {\cos {{3\pi } \over 8}}}\left( -{\cos {\pi \over 8} + i\sin {\pi \over 8}} \right) = {1 \over {\cos {{3\pi } \over 8}}}\left( {\cos {{7\pi } \over 8} + i\sin {{7\pi } \over 8}} \right)\)
\(c)\,\,1 - \cos \varphi - i\sin \varphi = 2\sin^2 {\varphi \over 2} - 2i\sin {\varphi \over 2}\cos {\varphi \over 2} = 2\sin {\varphi \over 2}\left[ {\sin {\varphi \over 2} - i\cos {\varphi \over 2}} \right]\)
Khi \(\sin {\varphi \over 2} > 0\) thì \(\,1 - \cos \varphi - i\sin \varphi = \left( {2\sin {\varphi \over 2}} \right)\left[ {\cos \left( {{\varphi \over 2} - {\pi \over 2}} \right) +i\sin\left( {{\varphi \over 2} - {\pi \over 2}} \right)} \right]\) là dạng lượng giác cần tìm.
Khi \(\sin {\varphi \over 2} < 0\) thì \(\,1 - \cos \varphi - i\sin \varphi = \left( { - 2\sin {\varphi \over 2}} \right)\left[ {\cos \left( {{\varphi \over 2} + {\pi \over 2}} \right) + i\sin \left( {{\varphi \over 2} + {\pi \over 2}} \right)} \right]\) là dạng lượng giác cần tìm.
Còn khi \(\sin {\varphi \over 2} = 0\) thì \(\,\,1 - \cos \varphi - i\sin \varphi = 0 = 0\left( {\cos \alpha + i\sin \alpha } \right)\,\,(\alpha \in\mathbb R\)tùy ý).
