Question

tg(a/2) - ctg(a/2)
———————— черта дроби
tg(a/2) + ctg(a/2)

Answer 1

$\frac{ \tan( \frac{a}{2} ) - \cot( \frac{a}{2} ) }{ \tan( \frac{a}{2} ) + \cot( \frac{a}{2} ) } = \frac{ \frac{ \sin( \frac{a}{2} ) }{ \cos( \frac{a}{2} ) } - \frac{ \cos( \frac{a}{2} ) }{ \sin( \frac{a}{2} ) } }{ \frac{ \sin( \frac{a}{2} ) }{ \cos( \frac{a}{2} ) } + \frac{ \cos( \frac{a}{2} ) }{ \sin( \frac{a}{2} ) } } = \\ = \frac{ \frac{ \sin ^{2} ( \frac{a}{2} ) - \cos ^{2} ( \frac{a}{2} ) }{ \cos( \frac{a}{2} ) \sin( \frac{a}{2} ) } }{ \frac{ \sin ^{2} ( \frac{a}{2} ) + \cos( \frac{a}{2} ) }{ \cos( \frac{a}{2} ) \sin( \frac{a}{2} ) } } = \frac{( \sin ^{2} ( \frac{a}{2} ) - \cos ^{2} ( \frac{a}{2} ))( \cos( \frac{a}{2} ) \sin( \frac{a}{2} )) }{ \cos( \frac{a}{2} ) \sin( \frac{a}{2} ) } = \\ = \sin ^{2} ( \frac{a}{2} ) - \cos ^{2} ( \frac{a}{2} ) = - ( \cos ^{2} ( \frac{a}{2} ) - \sin( \frac{a}{2} ) ) = - \cos(a)$