Example 6 Find the smallest positive period of the following functions:
(1) $y=\cos ^{1} x+\sin ^{1} x$;
(2) $y=2 \cos x \sin \left(x+\frac{\pi}{3}\right)-\sqrt{3} \cdot \sin ^{2} x+\sin x \cdot \cos x$.
Analyzing these two functions, both sums and products are involved, so we cannot directly apply formulas. Using the definition is also relatively difficult. When encountering such problems, the first step is to perform equivalent transformations to convert the known functions into expressions containing only one trigonometric function, and then use formulas to find the smallest positive period.

Solution
$$
\text{(1)} \begin{aligned}
y & =\left(\cos ^{2} x+\sin ^{2} x\right)^{2}-2 \cos ^{2} x \sin ^{2} x \\
& =1-\frac{1}{2} \sin ^{2} 2 x \\
& =1-\frac{1}{4}(1-\cos 4 x) .
\end{aligned}
$$

Thus, the smallest positive period \( T=\frac{2 \pi}{4}=\frac{\pi}{2} \).
(2)
$$
\begin{aligned}
y & =2 \cos x\left(\frac{1}{2} \sin x+\frac{\sqrt{3}}{2} \cos x\right)-\sqrt{3} \sin ^{2} x+\sin x \cos x \\
& =\sin 2 x+\sqrt{3} \cos 2 x=2 \sin \left(2 x+\frac{\pi}{3}\right),
\end{aligned}
$$

Thus, the smallest positive period \( T=\pi \).