Example 1-(2007 National Competition Henan Preliminary Question) Given $7 \sin \alpha+24 \cos \alpha=25$, find the value of $\tan \alpha$.
Analysis: From the known equation and the Cauchy-Schwarz inequality, we get 
\[1 \cdot \left(7^{2} + 24^{2}\right) = \left(\sin ^{2} \alpha + \cos ^{2} \alpha\right) \left(7^{2} + 24^{2}\right) \geqslant (7 \sin \alpha + 24 \cos \alpha)^{2} = 25^{2}.\]
The equality in the above equation holds if and only if \(\frac{\sin \alpha}{7} = \frac{\cos \alpha}{24}\), which implies \(\tan \alpha = \frac{7}{24}\).