Example 4 Solve the inequality $5 \sin ^{5} x+3 \sin ^{3} x>5 \cos ^{5} x+3 \cos ^{3} x$.

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Let $f(x)=5 x^{5}+3 x^{3}$, then the original inequality is $f(\sin x)>f(\cos x)$. Since $f(x)$ is a monotonically increasing function on $\mathbf{R}$, we have $\sin x>\cos x$.

Solving it, we get $2 k \pi+\frac{\pi}{4}<x<2 k \pi+\frac{5 \pi}{4}, k \in \mathbf{Z}$.