How to prove this series converges (possibly by modifying the Dirichlet test)?

Let me help you with this one.

We can rewrite the following expression as follows: $$ \sum_{n=1}^\infty {\sin(n)}\cdot(\sin(n)+n^{1/2})^{-1} \tag{1} $$ then we can take $n^{1/2}$ out and write it like $$ \sum_{n=1}^\infty {\frac{\sin(n)}{n^{1/2}}}\cdot\left (1+\frac{\sin(n)}{n^{1/2}}\right)^{-1}\tag{2} $$ now we can apply the taylor series like $$ \sum_{n=1}^\infty {\frac{\sin(n)}{n^{1/2}}}\cdot\left(1-\frac{\sin(n)}{n^{1/2}} + O\left(\frac{1}{n}\right)\right)\tag{3}$$

from here you can solve it, first one converges by Dirichlet, second one diverges, third one converges.