Question

Prove that 1-cos a + sin a / sin a + cos a -1 = 1+ sin a / cos a

Answer 1

Answer:

( 1 - cos a + sin a ) / ( sin a + cos a - 1 )

= ( sin a + ( 1 - cos a ) ) / ( sin a - ( 1 - cos a ) )

[ multiply numerator and denominator by ( sin a + ( 1 - cos a ) ) ]

= ( sin a + ( 1 - cos a ) )² / ( sin² a - ( 1 - cos a )² )

= ( sin² a + ( 1 - cos a )² + 2 sin a ( 1 - cos a ) ) / ( sin² a - 1 - cos² a + 2 cos a )

= ( sin² a + 1 + cos² a - 2 cos a + 2 sin a - 2 sin a cos a ) / ( - 2 cos² a + 2 cos a )

= ( 2 - 2 cos a + 2 sin a - 2 sin a cos a ) / (  2 cos a ( 1 - cos a )  )

= ( 1 - cos a + sin a - sin a cos a ) / ( cos a ( 1 - cos a ) )

= ( 1 - cos a ) ( 1 + sin a ) / ( cos a ( 1 - cos a ) )

= ( 1 + sin a ) / cos a

( Actually, the last step is only valid if 1 - cos a ≠ 0, so cos a ≠ 1. )