the correct question is
Given cos theta=4/9 and csc theta < 0 find sin theta and tan theta
we know that
csc theta=1/sin theta
if csc theta < 0
then
sin theta < 0
we have that
cos theta=4/9
we know that
sin² theta+cos² theta=1
so
sin² theta=1-cos² theta-----> 1-(4/9)²----> 1-(16/81)----> 65/81
sin theta=-√(65/81)---->-√65/9
the answer Part a) is
sin theta=-√65/9
Part b) find tan theta
tan theta=sin theta/cos theta
tan theta=(-√65/9)/(4/9)-----> tan theta=-√65/4
the answer part b) is
tan theta=-√65/4