Hey everyone! Today, we're diving deep into the world of trigonometry to simplify a specific function. You've probably encountered functions that look a bit messy, and you're wondering, "Can this be made simpler?" The answer is almost always yes! We're going to break down the function f(x) = cos x sin x / cos x and show you how to streamline it. This might seem straightforward, but understanding the steps involved is crucial for tackling more complex trigonometric problems down the line. So, buckle up, guys, because we're about to simplify this expression step-by-step.

Understanding the Basics of the Function

First off, let's talk about the function itself: f(x) = cos x sin x / cos x. What we have here is a fraction where the numerator is the product of the cosine of x and the sine of x, and the denominator is simply the cosine of x. When we're simplifying trigonometric functions, our main goal is to reduce the number of terms, eliminate redundancies, and express the function in its most basic form. This makes it easier to analyze, graph, and use in further calculations. Think of it like decluttering your workspace – the cleaner it is, the more efficiently you can work.

Now, the key thing to spot in f(x) = cos x sin x / cos x is that cos x appears in both the numerator and the denominator. This is a big clue that simplification is possible. However, there's a critical condition we need to keep in mind when dealing with fractions, especially in trigonometry: we cannot divide by zero. Therefore, for this function to be defined, the denominator, cos x, must not be equal to zero. We'll explore what this means for the domain of our function later on, but for now, let's focus on the algebraic simplification.

The Simplification Process: Step-by-Step

Alright, let's get down to business with simplifying f(x) = cos x sin x / cos x. The most obvious step is to cancel out the common term, which is cos x, from both the numerator and the denominator. It's just like if you had (2 * 3) / 2 – you can cancel the 2s and are left with 3. So, in our case:

f(x) = (cos x * sin x) / cos x

If we cancel out the cos x term, we are left with:

f(x) = sin x

And just like that, our complex-looking function has been reduced to the much simpler sin x! Pretty neat, right? This is the beauty of algebraic manipulation in trigonometry. It transforms complicated expressions into easily manageable ones. However, and this is a super important point, while the expression simplifies to sin x, we must acknowledge the initial restriction. The original function f(x) = cos x sin x / cos x is undefined when cos x = 0. The function g(x) = sin x, on the other hand, is defined for all real numbers. Therefore, the simplified function f(x) = sin x is only equivalent to the original function f(x) = cos x sin x / cos x for values of x where cos x is not equal to 0.

This distinction is vital in mathematics. When we simplify an expression, we often carry over the domain restrictions of the original expression to the simplified one. So, while the algebraic simplification leads us to sin x, it's crucial to remember that the original function had limitations. These limitations occur when cos x = 0. The cosine function equals zero at odd multiples of pi/2. That is, when x = (π/2) + nπ, where 'n' is any integer (..., -2, -1, 0, 1, 2, ...). So, our simplified function f(x) = sin x is valid for all x except x = (π/2) + nπ.

Domain Restrictions: The Unseen Rules

Let's really hammer home the point about domain restrictions because, guys, this is where a lot of people can trip up. The original function was f(x) = cos x sin x / cos x. The golden rule of any fraction is that the denominator cannot be zero. If the denominator is zero, the expression is undefined. In our case, the denominator is cos x. So, we must have cos x ≠ 0.

When does cos x = 0? This happens at specific points on the unit circle. Remember your unit circle values? Cosine is the x-coordinate. The x-coordinate is zero at the top (π/2) and bottom (3π/2) of the circle, and these points repeat every full rotation (2π). So, the values of x where cos x = 0 are: x = π/2, x = 3π/2, x = 5π/2, x = -π/2, and so on. A more concise way to express all these points is using the formula: x = (π/2) + nπ, where 'n' is any integer (..., -2, -1, 0, 1, 2, ...).

So, the simplified function is f(x) = sin x, but its domain is restricted. The domain of f(x) is all real numbers except for x = (π/2) + nπ, where 'n' is an integer. This means if you were to graph the original function f(x) = cos x sin x / cos x, it would look exactly like the graph of y = sin x, but with tiny holes at all the points where cos x = 0. These are called removable discontinuities.

Why is this important? Well, imagine you're using this function in a larger equation or a physics problem. If you plug in a value of x where cos x = 0, you'd get an error because the original function simply isn't defined there. The simplified form sin x might give you a value (like sin(π/2) = 1), but that value is not a valid output for the original function. Always, always, always consider the domain of the original function after simplification!

Why is This Simplification Useful?

Now you might be asking, "Why go through all this trouble?" Simplifying trigonometric functions like f(x) = cos x sin x / cos x is fundamental for several reasons, guys. Firstly, it makes analysis much easier. If you need to find the roots of the function, determine its behavior, or sketch its graph, working with sin x is infinitely simpler than working with (cos x sin x) / cos x. You can immediately recall the properties of the sine function – its amplitude, period, and range – without any extra effort.

Secondly, this simplification is a building block for more complex problems. In calculus, for instance, you'll need to differentiate or integrate trigonometric functions. Simplifying them beforehand can make these operations significantly less daunting. Imagine trying to apply the quotient rule to (cos x sin x) / cos x! It would be a nightmare. But if you simplify it to sin x first, the derivative is simply cos x, and the integral is -cos x + C. Huge difference!

Furthermore, understanding these simplifications helps in identifying identities. Trigonometric identities are equations that are true for all values of the variables for which both sides of the equation are defined. Recognizing that (cos x sin x) / cos x = sin x (under the domain restriction) is a form of using an identity. Mastering these basic manipulations builds a strong foundation for understanding and applying more advanced trigonometric identities, which are essential in fields like engineering, physics, and computer graphics.

It's also about efficiency. In any computational task involving these functions, using the simplified form requires fewer calculations, leading to faster processing times and reduced chances of numerical errors. So, while it might seem like a small step, the ability to simplify trigonometric expressions effectively is a powerful tool in your mathematical arsenal.

Conclusion: The Power of Simplification

So there you have it, folks! We took the function f(x) = cos x sin x / cos x, and through a simple process of cancellation, we arrived at f(x) = sin x. Remember, the key was spotting the common factor cos x in the numerator and denominator. However, we absolutely must carry over the domain restriction from the original function. The original function is undefined wherever cos x = 0, which occurs at x = (π/2) + nπ. Therefore, the simplified function f(x) = sin x is equivalent to the original f(x) = cos x sin x / cos x for all real numbers x except x = (π/2) + nπ.

This exercise highlights a fundamental principle in mathematics: always consider the domain of the original function. Simplification makes things easier, but it shouldn't erase the inherent limitations of the initial expression. By understanding and applying these principles, you're not just solving one problem; you're building a deeper understanding of how functions work and how to manipulate them effectively. Keep practicing these kinds of simplifications, and you'll become a trigonometry whiz in no time! Go forth and simplify!