Prove the equation

Question

csc / (1 + csc) = (1 - sin) / cos^2

Answer 1

Start with the left side of the equation and replace csc with (1/sin):

 

   csc    

1 + csc

 

    1

   sin      

1 +  1 

      sin

 

Multiply the top and the bottom of the fraction by sin; the terms with sin in the denominator will cancel because (1/sin)*sin = 1

 

    1    

1 + sin

 

Multiply the top and bottom of the fraction by the conjugate of 1 + sin; The conjugate is 1-sin.   You'll need to foil the (1 + sin)(1-sin):

 

    1       * (1-sin)

1 + sin     (1-sin)

 

       1 - sin           

1 - sin + sin - sin²

 

The -sin and +sin in the denominator cancel each other.

 

1 - sin

1 - sin²

 

Remember that cos² + sin² = 1.  If we subtract sin² from both sides of this equation, you get cos² = 1 - sin².  You can then substitute cos² into the denominator of your proof in place of 1 - sin²:

 

1 - sin

 cos²

 

And you're done!