solve each equation on the internal 0 is greater than equal to theta less than 2pie.
10) (cot theta +1)(csc theta -1/2) = 0
$\displaystyle (\cot{\theta} + 1)(\csc{\theta} - \frac{1}{2}) = 0$
Solutions will be when $\displaystyle \cot{\theta} = -1$ and when $\displaystyle \csc{\theta} = \frac{1}{2}$.
HOWEVER if $\displaystyle \csc{\theta} = \frac{1}{2}$, this implies $\displaystyle \sin{\theta} = 2$ which is invalid.
So the only solutions are found by solving $\displaystyle \tan{\theta} = -1$ (because if $\displaystyle \cot{\theta} = -1$ then so does $\displaystyle \tan{\theta}$)
Hint: $\displaystyle \tan \frac{3 \pi}{4} = -1$. Now use either the known period of $\displaystyle \tan \theta$ or the symmetry of the unit circle to find the second value of $\displaystyle \theta$.
You're expected to know these things. If you don't, then you should be reviewing earlier work on how to solve basic trig equations involving sin, cos and tan instead of attempting to solve the equations you've been posting.
Right, you know that $\displaystyle \cot{\theta} = -1$, and you should know that $\displaystyle \tan{\theta} = \frac{1}{\cot{\theta}}$. So in this case $\displaystyle \tan{\theta} = \frac{1}{-1} = -1$
You should know how to find the values for $\displaystyle \theta$ in the interval $\displaystyle 0 \leq \theta \leq 2\pi$ that satisfy this equation.
You should have $\displaystyle \theta = \left\{ \frac{3\pi}{4},~ \frac{7\pi}{4} \right\}$