100 points! only 29 and 31 rewrite each expression eliminating fractions. simplified answers may be left in terms of any of the six trigonometric functions.

29.  sec w csc w - sec^2 w

31.  - tan ^2 k (sin k +1)

Step-by-step explanation:

29.  (cot w -1)/ (1- sin^2 w)

We know that cot w = cos w/ sin w

and 1 - sin^2 w = cos ^2 w

Replace these identities into the expression

(cos w/ sin w -1)

cos ^ 2 w    

Get a common denominator on top of sin w

(cos w/ sin w -sin w/sin w)

cos ^ 2 w    

(cos w - sinw)/ sin w

cos ^ 2 w

(cos w - sinw)

cos ^ 2 w  sin w

Split into 2 terms

cos w                            sin w      

  -  

cos ^ 2 w  sin w             cos ^2 w  sin w

Cancel cos w in the first term and sin w in the second term

1                                       1    

  -  

cos  w  sin w             cos ^2 w  

We know that 1/cos = sec  and 1/ sin = csc

sec w csc w - sec^2 w

31.    sin k/ (1 - csc k)

We know that csc k is 1/ sin k

sin k

1 - 1/ sin k

Get a common denominator for the bottom

sin k

sin k/ sin k - 1/ sin k

sin k

(sin k-1)/ sin k

Multiply by sin k/sin k

sin k * sin k

(sin k-1)/ sin k  * sin k

sin k * sin k

(sin k-1)

sin^2 k

(sin k-1)

Multiply by (sin k +1)/ (sin k +1)  so we can get rid of the denominator

sin^2 k  (sin k +1)

(sin k-1) (sin k +1)

Foiling out the denominator, we get (sin^2 k-1)

sin^2 k  (sin k +1)

(sin^2 k-1)

Factoring out a -1 from the denominator

sin^2 k  (sin k +1)

-1 (1 - sin^2 k)

1 - sin ^2 k = cos ^ k

sin^2 k  (sin k +1)

-1 (cos ^2 k)

sin ^2k/ cos ^2 k = tan ^2 k

- tan ^2 k (sin k +1)

no this is a nonlinear negative