If-z-cos-isin-prove-that-1-z-2-1-z-2-itan-

Question Number 143097 by ZiYangLee last updated on 10/Jun/21

If z = \cos θ + i \sin θ,

prove that \frac{1 - z^{2}}{1 + z^{2}} = - i \tan θ

Answered by MJS_new last updated on 10/Jun/21

\frac{1 - {(c + is)}^{2}}{1 + {(c + is)}^{2}} = \frac{1 - c^{2} + s^{2} - 2 ics}{1 - s^{2} + c^{2} + 2 ics} = \frac{s^{2} - ics}{c^{2} + ics} =

= \frac{s (s - ic)}{c (c + is)} = \frac{- is z}{c z} = - i \tan θ

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