Question

Buktikan bahwa [tex] \frac{ \cot( \alpha ) - \tan( \alpha ) }{ \cot( \alpha ) - 1 } = 1 + \tan( \alpha ) [/tex]

anehnyaa org ini​

Answer 1

Verified answer

Jawaban:

[tex] \scriptsize{\frac{ \cot( \alpha ) - \tan( \alpha ) }{ \cot( \alpha ) - 1 } = 1 + \tan( \alpha )}\: \: \: \: \: \normalsize\rm{...(terbukti)}[/tex]

Penjelasan dengan langkah-langkah:

[tex]\begin{aligned} \tiny\frac{ \cot( \alpha ) - \tan( \alpha ) }{ \cot( \alpha ) - 1 } & =\tiny \frac{ \frac{1}{ \tan( \alpha ) } - \tan( \alpha ) }{ \frac{1}{ \tan( \alpha ) } - 1 } \\ & =\footnotesize\frac{ \frac{ \cos( \alpha ) }{ \sin( \alpha ) } - \frac{ \sin( \alpha ) }{ \cos( \alpha ) } }{ \frac{ \cos( \alpha ) }{ \sin( \alpha ) } - \frac{ \sin( \alpha ) }{ \sin( \alpha ) } } \\ & =\footnotesize \frac{ \frac{{ \cos}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } - \frac{{ \sin}^{2} ( \alpha ) }{ \sin( \alpha ) \cos( \alpha )} }{ \frac{ \cos( \alpha ) - \sin( \alpha ) }{ \sin( \alpha ) } } \\ & =\footnotesize \frac{ \frac{{ \cos}^{2} ( \alpha )- { \sin }^{2}( \alpha ) }{ \sin( \alpha ) \cos( \alpha ) } }{ \frac{ \cos( \alpha ) - \sin( \alpha ) }{ \sin( \alpha ) } } \\ & =\tiny \frac{{ \cos}^{2} ( \alpha )- { \sin }^{2}( \alpha ) ( \sin( \alpha )) }{ \sin( \alpha ) \cos( \alpha ) ( \cos( \alpha) - \sin( \alpha ) ) } \\ & = \tiny\frac{( \cos( \alpha ) - \sin( \alpha ) )( \cos( \alpha ) + \sin( \alpha ) )( \sin( \alpha )) }{ \sin( \alpha ) \cos( \alpha ) ( \cos( \alpha) - \sin( \alpha ) ) } \\ & =\tiny \frac{ \cos( \alpha ) + \sin( \alpha ) }{ \cos( \alpha ) }\\ & = \tiny \frac{ \cos( \alpha ) }{ \cos( \alpha ) }+ \frac{ \sin( \alpha ) }{ \cos( \alpha ) } \\ & = \footnotesize1 + \tan( \alpha )\: \: \: \: \: \tiny\rm{...(terbukti)} \end{aligned}[/tex]