Question

cosa-cosasina/cosa+cosasina

Answer 1

Answer:

$\frac{ \cos( \alpha ) - \cos( \alpha ) \sin( \alpha ) }{ \cos( \alpha ) + \cos( \alpha ) \sin( \alpha ) } = \frac{ \cos( \alpha ) - \cos( \alpha ) \sin( \alpha ) }{ \cos( \alpha ) + \cos( \alpha ) \sin( \alpha ) } \times \frac{ \cos( \alpha ) - \cos( \alpha ) \sin( \alpha ) }{ \cos( \alpha ) - \cos( \alpha ) \sin( \alpha ) } = \frac{ (\cos \alpha - \cos \alpha \sin \alpha ) {}^{2} }{ \cos {}^{2} ( \alpha ) - \cos {}^{2} ( \alpha ) \sin {}^{2} ( \alpha ) } =$

$\frac{ \cos {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) \sin {}^{2} ( \alpha ) - 2 \cos {}^{2} ( \alpha ) \sin( \alpha ) }{ \cos {}^{2} ( \alpha ) - \cos {}^{2} ( \alpha ) \sin {}^{2} ( \alpha ) } =$

$\frac{ \cos {}^{2} ( \alpha ) (1 + \sin {}^{2} ( \alpha ) - 2 \sin( \alpha )) }{ \cos {}^{2} ( \alpha )(1 - \sin {}^{2} ( \alpha ) )} =$

$\frac{(1 - \sin \alpha) {}^{2} }{ \cos {}^{2} \alpha } = ( \frac{1}{ \cos \alpha } - \frac{ \sin \alpha }{ \cos \alpha } ) {}^{2} = ( \sec \alpha - \tan \alpha ) {}^{2}$