How do you simplify csc^2x secx/sec^2x+csc^2x?

Geno and I didn't make any errors (that I know of)......we just interpreted this differently......the poster can see which matches his question.....

csc²(x) · sec(x) / sec²(x)  +  csc²(x)

=  csc²(x) · [ sec(x) / sec²(x) ]  +  csc²(x)                           

=  csc²(x) · [ 1 /sec(x) ]  +  csc²(x)                        <---   simplify:  sec(x) / sec²(x)

=  csc²(x) · ( cos(x) )  +  csc²(x)                            <---    cos(x)  =  1 / sec(x)

=  csc²(x) · [ cos(x) + 1 ]                                        <---    factor

=  [ 1 / sin²(x) ] · [ cos(x) + 1 ]                              <---    csc²(x)  =  1 / sin²(x)

=  [ cos(x) + 1 ] / sin²(x)

=  [ cos(x) + 1 ] / [ 1 - cos²(x) ]                             <---    sin²(x)  =  1 - cos²(x)

=  [ cos(x) + 1 ] / [ ( 1 - cos(x) )·( 1 - cos(x) ) ]      <---    factor

=  1 / [ 1 - cos(x) ]                 <--- provided that sec²(x) ≠ 0, cos(x) ≠ 0

I think this might be  [csc^2x secx] / [sec^2x+csc^2x ].....if so, convert to sines and  cosines

[ 1 / sin^2x * 1 / cosx ] / [ 1 / cos^2x  + 1 / sin^2x ]     simplify the denominator

[ 1 / sin^2x * 1 / cosx ] / [(sin^2x + cos^2x) / (cos^2x sin^2x) ] =

[ 1 /(sin^2x * cosx)] / [1 /( cos^2x * sin^2x) ] =

( cos^2x * sin^2x) ] / (sin^2x * cosx) =

cosx

2 answers WOW

2 for the price of one - arn't you the lucky anon.

Actually it is kind of good.  You should be able to work out where the error lies.  :)

It is an excellent learning technique. 

okay - I didn't actually look