Hey guys! Ever stumbled upon a trigonometric expression that looks a bit intimidating? Well, today, we're going to break down a seemingly complex function into something super simple and easy to understand. Our mission? To simplify the expression f(x) = (cos x * sin x) / cos x. Sounds like a mouthful, right? Don't worry; we'll take it step by step. So, grab your calculators (though you might not need them!), and let's dive in!

Understanding the Initial Expression

First, let's get cozy with our starting point: f(x) = (cos x * sin x) / cos x. At first glance, it might seem like we're juggling a bunch of trigonometric identities and complex rules. But trust me, it's much simpler than it looks. The key here is to recognize the components we're working with. We have cos x, which represents the cosine function of the angle x, and sin x, which represents the sine function of the angle x. These are fundamental trigonometric functions that relate angles of a right triangle to the ratios of its sides. When we see these functions combined in an expression like this, we should immediately start thinking about potential simplifications. In our case, we have cos x multiplied by sin x in the numerator, and then the entire product is divided by cos x. This setup hints at a possible cancellation, which is the first thing we should investigate when trying to simplify any algebraic expression. Always remember, identifying the basic elements and looking for opportunities to simplify early on can save you a lot of headaches later. Trigonometry can be tricky, but breaking it down into manageable parts makes it much less daunting.

Identifying Opportunities for Simplification

Now, let's pinpoint where we can make things easier. Remember those good old algebra rules? They're our best friends here. We see that cos x appears both in the numerator (as a factor) and in the denominator. This is a classic setup for simplification through cancellation. Think of it like having the number 5 multiplied on top and bottom of a fraction – you can cancel them out! But before we jump the gun, we need to be a little cautious. We can only cancel out terms that are multiplied, not added or subtracted. In our expression, cos x is indeed multiplied by sin x in the numerator, and it stands alone in the denominator. So, we're good to go! This is a crucial step because incorrectly applying cancellation is a common mistake. Always double-check that the terms you're planning to cancel are factors in a product. Also, keep in mind that cos x cannot be zero. If cos x = 0, the original expression would be undefined due to division by zero. However, for the purpose of simplifying the expression where cos x is not zero, we can proceed with the cancellation. By carefully observing the structure of the expression and understanding the rules of algebra, we've identified a clear path to simplification.

Performing the Simplification

Alright, time for the magic! We're going to cancel out the cos x terms. So, (cos x * sin x) / cos x becomes sin x after cancellation. Boom! That's it! The cos x in the numerator and the cos x in the denominator simply eliminate each other, leaving us with just sin x. Isn't that neat? It's like watching a complicated puzzle piece slide perfectly into place. Now, it's important to state our simplified function clearly. After performing the simplification, we have: f(x) = sin x. This is our simplified expression. It's much cleaner and easier to work with than the original. Simplifying expressions like this is not just an academic exercise; it's a practical skill that makes further calculations and analysis much more manageable. Imagine trying to graph the original function versus graphing sin x – the latter is far easier! So, give yourself a pat on the back; you've just turned a potentially confusing expression into something beautifully simple.

State the Simplified Function

After simplifying the original function f(x) = (cos x * sin x) / cos x, we arrive at a much cleaner and concise form. By canceling out the cos x terms in both the numerator and the denominator, we are left with the simplified function: f(x) = sin x. This means that the original expression is equivalent to the sine function of x, provided that cos x is not equal to zero. The simplification not only makes the function easier to understand but also significantly reduces the effort required for further analysis, such as plotting the graph of the function, finding its derivatives, or integrating it. The simplified form allows us to quickly recognize the fundamental properties of the function, such as its periodicity, amplitude, and symmetry. Moreover, the process of simplification highlights the importance of identifying and canceling out common factors in mathematical expressions, a technique that is widely applicable in various fields of mathematics and science. Understanding and applying these simplification techniques is crucial for solving more complex problems and gaining deeper insights into mathematical relationships. So, remember, always look for opportunities to simplify; it can make a world of difference!

Important Considerations: When cos x = 0

Now, hold on a second! While we've successfully simplified the expression, there's a tiny little detail we need to address. Remember when we cancelled out cos x? Well, we can only do that if cos x is not equal to zero. Why, you ask? Because dividing by zero is a big no-no in the math world – it's undefined! So, what happens when cos x = 0? Let's think about it. The original function was f(x) = (cos x * sin x) / cos x. If cos x is indeed zero, then the denominator becomes zero, making the entire expression undefined. This means that our simplified function, f(x) = sin x, is equivalent to the original function everywhere except where cos x = 0. So, where does cos x equal zero? Well, cos x = 0 when x is equal to π/2 (90 degrees) plus any integer multiple of π (180 degrees). In mathematical terms, cos x = 0 when x = π/2 + nπ, where n is an integer. At these points, the original function is undefined, while our simplified function, sin x, is perfectly happy and defined. Therefore, when stating our simplified function, we need to add a little disclaimer: f(x) = sin x, for x ≠ π/2 + nπ, where n is an integer. This ensures that we're being mathematically accurate and not misleading anyone about the behavior of the original function. Always remember to consider these edge cases when simplifying expressions, especially when dealing with division!

Visualizing the Simplification

To really drive the point home, let's visualize what we've done. Imagine graphing both the original function, f(x) = (cos x * sin x) / cos x, and the simplified function, f(x) = sin x. What would you see? Well, for the most part, the two graphs would be identical. They would both oscillate up and down like a typical sine wave, ranging from -1 to 1. However, there would be a subtle difference. At the points where cos x = 0 (i.e., x = π/2 + nπ, where n is an integer), the graph of the original function would have vertical asymptotes. This means that the function would approach infinity (or negative infinity) as x gets closer and closer to these values. On the other hand, the graph of the simplified function, f(x) = sin x, would be smooth and continuous at these points, with a defined value of either 1 or -1. This visual representation highlights the importance of the condition we added to our simplified function: f(x) = sin x, for x ≠ π/2 + nπ, where n is an integer. It shows that while the simplified function is a good representation of the original function, it doesn't capture the undefined behavior at certain points. By visualizing the functions, we gain a deeper understanding of the simplification process and its limitations.

Conclusion

So, there you have it, folks! We've successfully simplified the expression f(x) = (cos x * sin x) / cos x to f(x) = sin x, with the crucial caveat that x ≠ π/2 + nπ, where n is an integer. We started by understanding the initial expression, identified opportunities for simplification through cancellation, performed the simplification, and then carefully considered the cases where cos x = 0. By following these steps, we not only arrived at a simpler expression but also gained a deeper understanding of the underlying mathematical principles. Remember, simplifying expressions is a valuable skill in mathematics and science, and it often involves a combination of algebraic manipulation, trigonometric identities, and careful consideration of potential pitfalls. So, keep practicing, keep exploring, and keep simplifying! You'll be amazed at how much easier complex problems become when you break them down into smaller, more manageable parts. And most importantly, don't be afraid to ask questions and seek help when you get stuck. Happy simplifying!