3. \( \cos ^{2} \theta\left(1+\tan ^{2} \theta\right) \)

Did you know that this expression is deeply connected to one of the fundamental identities in trigonometry? The identity \(1 + \tan^2 \theta = \sec^2 \theta\) shows that the expression simplifies nicely. Thus, you can rewrite \( \cos^2 \theta (1 + \tan^2 \theta) \) as \( \cos^2 \theta \sec^2 \theta\), and since \( \sec \theta = \frac{1}{\cos \theta}\), you end up with just \(1\). How handy is that? Now, let’s put this into real-world context! Trigonometric functions are not just for math class; they are used in fields like engineering, physics, and even computer graphics! Whenever you need to model waves, oscillations, or even projectors on a screen, these friendly trigonometric identities come to your rescue, making complex calculations much simpler and more manageable!