Example 4. Find (1) the principal period of $f(x)=\operatorname{tg} 2 x-\operatorname{ctg} 2 x$; (2) the principal period of $g(x)=12 \sin ^{2} 15 x+15 \cos ^{2} 12 x$.
(1) The conclusion that the main period of $f(x)=\tan 2x - \cot 2x$ is $\frac{\pi}{2}$ is incorrect. This is because the conclusions for sine and cosine functions cannot be directly applied to tangent and cotangent functions.
$$
\begin{array}{c}
\because f(x)=\tan 2x - \cot 2x = \frac{\sin 2x}{\cos 2x} - \frac{\cos 2x}{\sin 2x} = \frac{-\cos 4x}{\frac{1}{2} \sin 4x} = -2 \cot 4x, \\
\therefore \text{the main period of } f(x) \text{ is } \frac{\pi}{4}.
\end{array}
$$
$$
\begin{aligned}
& \text{(2) } \because g(x) = 12 \sin^2 15x + 15 \cos^2 12x \\
= & 12 \times \frac{1 - \cos 30x}{2} + 15 \times \frac{1 + \cos 24x}{2} \\
= & 5(1 - \cos 30x) + \frac{15}{2}(1 + \cos 24x),
\end{aligned}
$$

According to the conclusion of Example 3, the main period of $g(x)$ is $\frac{\pi}{3}$.
Trigonometric function expressions can often be transformed to find the main period of the function.