SOLUTION: using trigonometric identities to simplify the expression (csc theta)(sec theta - cos theta)

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Question 84048: using trigonometric identities to simplify the expression (csc theta)(sec theta - cos theta)
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
using trigonometric identities to simplify the expression
cscqsecq - cosq

Are you sure you copied it right? Are you sure it wasn't
supposed to be 

cscqsecq - cotq  ??

I'm going to assume it was supposed to be "cotq"
and not "cosq"; for otherwise it won't simplify at all!

cscqsecq - cotq

               1                1                cosq
Since cscq = ------ , secq = ------, and cotq = ------
              sinq            cosq               sinq

   1      1      cosq
------·------ - ------
 sinq   cosq     sinq

     1        cosq  
---------- - ------
 sinqcosq     sinq

Get an LCD of sinqcosq. Multiply the second
fraction top and bottom by cosq 

     1        cosq(cosq)  
---------- - ---------------
 sinqcosq     sinq(cosq)


     1        cos2q  
---------- - -----------
 sinqcosq     sinqcosq

Combine numerators over LCD:

 1 - cos2q  
------------
  sinqcosq 

Now since sin2q + cos2q = 1, then 1 - cos2q = sin2q,
so the numerator is just sin2q

  sin2q
----------
 sinqcosq

Then we can cancel the sinq in the
bottom into the sin2q in the top 
  
  sinq
  sin2q
----------
 sinqcosq
  1

Then all that's left is

 sinq
------  
 cosq
                  sinq 
and since tanq = ------
                  cosq

The final answer is

tanq

Edwin