Simplify Trig Expression: Find F(x) For Cot(-x)cos(-x)+sin(-x)

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Have you ever encountered a trigonometric expression that looks daunting at first glance? These expressions, filled with sines, cosines, and other trig functions, can seem complex. But don't worry! With a few key identities and simplification techniques, we can break them down into manageable parts. This article will guide you through the process of simplifying a specific trigonometric expression, cot⁡(−x)cos⁡(−x)+sin⁡(−x){\cot(-x) \cos(-x) + \sin(-x)}, and help you find the function f(x){f(x)} that makes the entire expression equal to −1f(x){-\frac{1}{f(x)}}. Let's dive in and unravel the mysteries of trigonometric simplification!

Understanding the Trigonometric Expression

Before we begin simplifying, it’s crucial to understand each component of the expression: cot⁡(−x)cos⁡(−x)+sin⁡(−x){\cot(-x) \cos(-x) + \sin(-x)}. This expression involves three fundamental trigonometric functions: cotangent (cot), cosine (cos), and sine (sin). The argument of each function is −x{-x}, which indicates a negative angle. To simplify this expression, we'll need to utilize trigonometric identities that deal with negative angles and the relationships between different trigonometric functions. The key here is to remember that trigonometric functions have specific properties regarding negative angles. For instance, cosine is an even function, meaning cos⁡(−x)=cos⁡(x){\cos(-x) = \cos(x)}, while sine and cotangent are odd functions, meaning sin⁡(−x)=−sin⁡(x){\sin(-x) = -\sin(x)} and cot⁡(−x)=−cot⁡(x){\cot(-x) = -\cot(x)}. Recognizing these properties is the first step towards simplifying the expression. Furthermore, we need to recall the definition of cotangent in terms of sine and cosine, which is cot⁡(x)=cos⁡(x)sin⁡(x){\cot(x) = \frac{\cos(x)}{\sin(x)}}. This substitution will allow us to express the entire expression in terms of sine and cosine, which is our ultimate goal. Understanding these fundamental concepts and trigonometric identities will pave the way for a smoother simplification process.

Applying Trigonometric Identities

Now that we understand the components, let’s apply some key trigonometric identities to simplify the expression. Remember, our goal is to express everything in terms of sine and cosine. We start with the given expression: cot⁡(−x)cos⁡(−x)+sin⁡(−x){\cot(-x) \cos(-x) + \sin(-x)}. First, we address the negative angles. Using the identities for odd and even functions, we know that cot⁡(−x)=−cot⁡(x){\cot(-x) = -\cot(x)}, cos⁡(−x)=cos⁡(x){\cos(-x) = \cos(x)}, and sin⁡(−x)=−sin⁡(x){\sin(-x) = -\sin(x)}. Substituting these into the expression, we get:

−cot⁡(x)cos⁡(x)−sin⁡(x){-\cot(x) \cos(x) - \sin(x)}

Next, we need to express cotangent in terms of sine and cosine. We know that cot⁡(x)=cos⁡(x)sin⁡(x){\cot(x) = \frac{\cos(x)}{\sin(x)}}. Replacing cot⁡(x){\cot(x)} with this fraction, the expression becomes:

−cos⁡(x)sin⁡(x)cos⁡(x)−sin⁡(x){-\frac{\cos(x)}{\sin(x)} \cos(x) - \sin(x)}

This substitution is crucial because it brings us closer to our goal of having the expression solely in terms of sine and cosine. Now, we can simplify the first term by multiplying the cosines:

−cos⁡2(x)sin⁡(x)−sin⁡(x){-\frac{\cos^2(x)}{\sin(x)} - \sin(x)}

At this point, we have successfully applied trigonometric identities to transform the original expression into a form that involves only sine and cosine functions. This step demonstrates the power of using identities to manipulate and simplify trigonometric expressions, making them easier to work with and understand.

Simplifying the Expression

With the expression now in terms of sine and cosine, let's further simplify it. We have:

−cos⁡2(x)sin⁡(x)−sin⁡(x){-\frac{\cos^2(x)}{\sin(x)} - \sin(x)}

To combine these two terms, we need a common denominator. The common denominator here is sin⁡(x){\sin(x)}. We rewrite the second term with this denominator:

−cos⁡2(x)sin⁡(x)−sin⁡2(x)sin⁡(x){-\frac{\cos^2(x)}{\sin(x)} - \frac{\sin^2(x)}{\sin(x)}}

Now that we have a common denominator, we can combine the fractions:

−cos⁡2(x)−sin⁡2(x)sin⁡(x){\frac{-\cos^2(x) - \sin^2(x)}{\sin(x)}}

Notice that we can factor out a -1 from the numerator:

−(cos⁡2(x)+sin⁡2(x))sin⁡(x){\frac{-(\cos^2(x) + \sin^2(x))}{\sin(x)}}

This is where another fundamental trigonometric identity comes into play: the Pythagorean identity, which states that cos⁡2(x)+sin⁡2(x)=1{\cos^2(x) + \sin^2(x) = 1}. Substituting this into our expression, we get:

−1sin⁡(x){\frac{-1}{\sin(x)}}

This simplification is significant because it reduces the entire expression to a single term. The expression is now in its simplest form in terms of sine and cosine, and we can easily identify the function f(x){f(x)} that satisfies the given condition.

Finding f(x)

Our final step is to find the function f(x){f(x)} such that the simplified expression equals −1f(x){-\frac{1}{f(x)}}. We've simplified the original expression to:

−1sin⁡(x){\frac{-1}{\sin(x)}}

We are given that this simplified expression is equal to −1f(x){-\frac{1}{f(x)}}. Therefore, we have:

−1f(x)=−1sin⁡(x){-\frac{1}{f(x)} = \frac{-1}{\sin(x)}}

By comparing both sides of the equation, it's clear that:

f(x)=sin⁡(x){f(x) = \sin(x)}

This concludes our simplification process. We have successfully simplified the given trigonometric expression and found the function f(x){f(x)} that satisfies the given condition. The function f(x){f(x)} is simply the sine function, sin⁡(x){\sin(x)}. This exercise highlights the importance of trigonometric identities and algebraic manipulation in simplifying complex expressions. Understanding these concepts is crucial for solving a wide range of problems in trigonometry and calculus.

Conclusion

In this article, we tackled the challenge of simplifying the trigonometric expression cot⁡(−x)cos⁡(−x)+sin⁡(−x){\cot(-x) \cos(-x) + \sin(-x)} and expressing it in terms of sine and cosine. By applying fundamental trigonometric identities, such as those for negative angles and the Pythagorean identity, we were able to reduce the expression to its simplest form: −1sin⁡(x){\frac{-1}{\sin(x)}}. We then determined that the function f(x){f(x)} that satisfies the equation −1f(x)=−1sin⁡(x){-\frac{1}{f(x)} = \frac{-1}{\sin(x)}} is f(x)=sin⁡(x){f(x) = \sin(x)}. This process underscores the importance of mastering trigonometric identities and algebraic simplification techniques. These skills are not only essential for solving mathematical problems but also for understanding various concepts in physics, engineering, and other scientific fields. Remember, practice is key to mastering these concepts. The more you work with trigonometric expressions, the more comfortable and confident you'll become in simplifying them.

For further exploration of trigonometric identities and functions, you might find the resources at Khan Academy's Trigonometry section helpful.