Expert: Sherry Wallin - 1/12/2009

Question
How do you prove this statement?
cosx/(secx-1)-cosx/tan^2x=cot^2x

Answer
Hi Mary~
    To prove this statement, take one side or the other and try to derive the remaining one. I chose to take the left and arrive at the right in this way taking one term at a time:

(secx +1)cosx            
------------------        
(secx +1)(secx-1)   

I chose to multiply by (secx + 1) because it's the conjugate of (secx-1)and it will just leave me with sec^2 x -1 which is the tan^2x
  
                          
= (secx +1)cosx
 ----------------    
      tan^2x         

notice the denominator of the first term is now the same as the
denominator of the 2nd term which is also another good reason to proceed as I did

=[(secx +1)cosx]/tan^2x-cosx/tan^2x =[(secx +1)cosx-cosx]/tan^2x

= cosx(secx+1-1)/tan^2x = cosx*secx/tan^2x
     
Note that cos x and sec x are reciprocals (inverses of each other which means their product is 1).

cosx*secx = cosx(1/cosx)= 1

so we have 1/tan^2x = cot^2x

There you have it, the right hand side