Answer:
29. sec w csc w - sec^2 w
31. - tan ^2 k (sin k +1)
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)