A foundational theorem of mathematical analysis states: Absolute convergence implies convergence.

That is, if the series of absolute values $\sum_{n=1}^\infty |a_n|$ converges, then the series $\sum_{n=1}^\infty a_n$ must necessarily converge.
Proof follows from Cauchy criterion: $|\sum_{k=m}^n a_k| \le \sum_{k=m}^n |a_k| \lt \epsilon$.

Consider the absolute value of the general term $a_n$:

Since $|\sin nx| \le 1$ for all real $x$ and all $n$:

The series $\sum_{n=1}^\infty \frac{1}{n^2}$ is a convergent p-series ($p = 2 \gt 1$).
By the direct comparison test (and Weierstrass M-test), $\sum_{n=1}^\infty |a_n|$ converges uniformly and absolutely for all $x \in \mathbb{R}$.

Given $c \gt 1$:
Since $c \gt 1 \gt 0$, $c$ is strictly positive.
Multiplying both sides of the inequality $1 \lt c$ by $c$ yields:

According to D'Alembert's Ratio Test for infinite series of real numbers:
Let $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$.
If $L \lt 1$, the series converges absolutely.
If $L \gt 1$, the series diverges.
If $L = 1$, the test is inconclusive.