$ \begin{array}{l} {{r}{ove}\hspace{0.33em}{the}\hspace{0.33em}{idenity}\hspace{0.33em}\tan\frac{\mathit{\theta}}{2}\mathrm{{=}}\frac{{1}\mathrm{{-}}\cos\mathit{\theta}}{\sin\mathit{\theta}}{\mathrm{.}}}\\ {\frac{{1}\mathrm{{-}}\cos\mathit{\theta}}{\sin\mathit{\theta}}\mathrm{{=}}\frac{{1}\mathrm{{-}}\cos{2}{\mathrm{(}}\frac{\mathit{\theta}}{2}{\mathrm{)}}}{\sin2\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}}}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{{1}\mathrm{{-}}{\mathrm{[}}{2}{\cos}^{2}{\mathrm{(}}\frac{\mathit{\theta}}{2}{\mathrm{)}}\mathrm{{-}}{1}{\mathrm{]}}}{2\sin\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}\cos\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}}}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{{2}{\mathrm{[}}{1}\mathrm{{-}}{\cos}^{2}{\mathrm{(}}\frac{\mathit{\theta}}{2}{\mathrm{)]}}}{2\sin\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}\cos\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}}}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{2\mathrm{[}{\sin}^{2}\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)]}}{2\sin\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}\cos\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}}}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{\sin\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}}{\cos\mathrm{(}\frac{\mathit{\theta}}{2}\mathrm{)}}}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\tan\frac{\mathit{\theta}}{2}}\\ {}\\ {NOTE}\\ {{sin}{2}\mathit{\theta}\mathrm{{=}}{2}{sin}\mathit{\theta}{cos}\mathit{\theta}}\\ {{cos}{2}\mathit{\theta}\mathrm{{=}}{2}{cos}^{2}\mathit{\theta}\mathrm{{-}}{1}}\\ {{tan}\mathit{\theta}\mathrm{{=}}\mathrm{\frac{{sin}\mathit{\theta}}{{cos}\mathit{\theta}}}} \end{array} $