Establish The Trigonometric Identity -cot(t) + Sin(t) / (1 - Cos(t)) = Csc(t)
Introduction
In the realm of trigonometry, establishing identities is a fundamental skill that allows us to manipulate and simplify complex expressions. These identities serve as building blocks for solving equations, proving theorems, and understanding the relationships between trigonometric functions. In this article, we will embark on a step-by-step journey to establish the trigonometric identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t). This identity showcases the interplay between cotangent, sine, cosine, and cosecant functions, providing a valuable exercise in trigonometric manipulation and simplification. By delving into this identity, we will not only strengthen our understanding of trigonometric functions but also hone our problem-solving abilities in mathematics.
Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined. They are used to simplify trigonometric expressions, solve trigonometric equations, and prove other trigonometric identities. The identity we are about to prove, -cot(t) + sin(t) / (1 - cos(t)) = csc(t), is a classic example that demonstrates how different trigonometric functions can be related and manipulated to achieve a simplified form. Understanding and mastering such identities is crucial for anyone delving deeper into trigonometry and calculus, where these identities are frequently used to solve more complex problems. This exploration will not only solidify your understanding of fundamental trigonometric concepts but also enhance your ability to approach and solve mathematical problems strategically.
This article aims to provide a comprehensive guide to establishing the given trigonometric identity. We will begin by expressing the left-hand side of the equation in terms of sine and cosine, which is a common strategy for simplifying trigonometric expressions. Then, we will combine the fractions using algebraic techniques, carefully manipulating the expression to reveal familiar trigonometric relationships. Finally, through a series of strategic steps, we will transform the left-hand side into the right-hand side, thus establishing the identity. Throughout this process, we will emphasize the underlying principles and techniques involved, ensuring that you not only understand the specific steps but also grasp the broader strategies for tackling trigonometric identities. By the end of this article, you will be equipped with the knowledge and skills to confidently tackle similar problems and deepen your understanding of trigonometric identities.
Step 1: Write the Left Side in Terms of Sine and Cosine
The initial step in establishing the identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t) is to express the left-hand side (LHS) entirely in terms of sine and cosine functions. This is a common and often effective strategy in simplifying trigonometric expressions, as it allows us to work with the fundamental building blocks of trigonometric functions. The cotangent function, cot(t), is defined as the ratio of cosine to sine, specifically cos(t) / sin(t). Similarly, the cosecant function, csc(t), which we aim to obtain on the right-hand side, is the reciprocal of the sine function, or 1 / sin(t). By converting all terms to sine and cosine, we create a common ground for algebraic manipulation and simplification.
Starting with the LHS, -cot(t) + sin(t) / (1 - cos(t)), we replace cot(t) with its equivalent expression cos(t) / sin(t). This substitution gives us - (cos(t) / sin(t)) + sin(t) / (1 - cos(t)). Now, the expression consists solely of sine and cosine functions, which sets the stage for combining the terms and simplifying the expression. This transformation is a crucial step, as it allows us to leverage algebraic techniques such as finding a common denominator and combining fractions, which are essential tools in simplifying complex trigonometric expressions. By expressing everything in terms of sine and cosine, we've effectively laid the groundwork for a more manageable algebraic manipulation.
This step highlights the importance of recognizing and utilizing fundamental trigonometric identities. The ability to express trigonometric functions in terms of sine and cosine is a cornerstone of trigonometric manipulation. It is a technique that appears frequently in various contexts, from solving trigonometric equations to proving more complex identities. By mastering this fundamental step, you gain a significant advantage in tackling a wide range of trigonometric problems. Furthermore, this initial transformation often reveals hidden structures and relationships within the expression, paving the way for subsequent simplification steps. Therefore, writing trigonometric expressions in terms of sine and cosine is not merely a mechanical step but a strategic decision that can significantly simplify the problem-solving process.
Step 2: Add the Fractions and Apply Discussion Category
Having expressed the left-hand side of the identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t) in terms of sine and cosine as - (cos(t) / sin(t)) + sin(t) / (1 - cos(t)), the next crucial step is to combine these fractions. To add or subtract fractions, a common denominator is required. In this case, the denominators are sin(t) and (1 - cos(t)), so the least common denominator (LCD) is simply their product, sin(t) * (1 - cos(t)). We must now rewrite each fraction with this common denominator.
The first fraction, - (cos(t) / sin(t)), needs to be multiplied by (1 - cos(t)) / (1 - cos(t)). This gives us -cos(t) * (1 - cos(t)) / (sin(t) * (1 - cos(t))). Similarly, the second fraction, sin(t) / (1 - cos(t)), needs to be multiplied by sin(t) / sin(t), resulting in sin^2(t) / (sin(t) * (1 - cos(t))). Now that both fractions have the same denominator, we can add them together. The resulting expression is (-cos(t) * (1 - cos(t)) + sin^2(t)) / (sin(t) * (1 - cos(t))). This combined fraction is a significant milestone in our simplification process.
Next, we need to simplify the numerator of the combined fraction. Expanding the term -cos(t) * (1 - cos(t)) gives us -cos(t) + cos^2(t). So, the numerator becomes -cos(t) + cos^2(t) + sin^2(t). Now, we can apply a fundamental trigonometric identity: sin^2(t) + cos^2(t) = 1. Substituting this identity into the numerator simplifies it further to -cos(t) + 1. Thus, the entire expression now looks like (1 - cos(t)) / (sin(t) * (1 - cos(t))). This simplification process showcases the power of combining algebraic manipulation with trigonometric identities.
The discussion category that this problem falls under is mathematics, specifically trigonometry. This problem involves the manipulation of trigonometric functions and the application of trigonometric identities to establish an equality. It is a classic example of the type of problems encountered in trigonometry courses and is essential for developing a strong foundation in the subject. Understanding how to manipulate trigonometric expressions and apply identities is crucial for solving more complex problems in calculus, physics, and other fields that rely on trigonometric principles. This problem highlights the importance of algebraic skills in simplifying trigonometric expressions, as well as the strategic application of fundamental trigonometric identities to achieve the desired result.
Step 3: Simplify the Expression
After combining the fractions and applying the Pythagorean identity, our expression has been simplified to (1 - cos(t)) / (sin(t) * (1 - cos(t))). The next logical step in establishing the identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t) is to further simplify this expression. We can observe that the factor (1 - cos(t)) appears in both the numerator and the denominator of the fraction. This allows us to cancel out this common factor, provided that (1 - cos(t)) is not equal to zero. The condition (1 - cos(t)) ≠ 0 is equivalent to cos(t) ≠ 1.
Assuming that cos(t) ≠ 1, we can cancel out the (1 - cos(t)) terms in the numerator and denominator. This simplification transforms the expression into 1 / sin(t). Now, we recognize that 1 / sin(t) is the definition of the cosecant function, csc(t). Therefore, we have successfully transformed the left-hand side of the original equation into csc(t), which is the right-hand side of the identity. This completes the proof of the identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t).
This simplification step underscores the importance of recognizing common factors in mathematical expressions. Cancelling common factors is a fundamental algebraic technique that can significantly simplify complex expressions. In this case, identifying and cancelling the (1 - cos(t)) term was crucial in reaching the final simplified form. Furthermore, this step highlights the connection between algebraic manipulation and trigonometric identities. The ability to combine these skills is essential for solving a wide range of trigonometric problems.
The condition cos(t) ≠ 1 is an important consideration. While we have shown that the identity holds true when cos(t) ≠ 1, we should also acknowledge this restriction. The identity is not defined when cos(t) = 1, as this would result in division by zero in the original expression. This emphasizes the importance of considering the domain of trigonometric functions and the restrictions that may arise due to division by zero or other mathematical singularities. The final simplified expression, csc(t), is also undefined when sin(t) = 0, which further restricts the domain of the identity.
Conclusion
In conclusion, we have successfully established the trigonometric identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t) through a series of logical steps and algebraic manipulations. We began by expressing the left-hand side of the equation in terms of sine and cosine functions, which is a common and effective strategy for simplifying trigonometric expressions. This initial transformation allowed us to work with the fundamental building blocks of trigonometry and set the stage for subsequent steps.
Next, we combined the fractions on the left-hand side by finding a common denominator and adding the numerators. This step involved careful algebraic manipulation and the application of the distributive property. After combining the fractions, we simplified the numerator by expanding terms and applying the Pythagorean identity, sin^2(t) + cos^2(t) = 1. This identity played a crucial role in simplifying the expression and revealing a common factor that could be cancelled out.
Finally, we cancelled the common factor (1 - cos(t)) from the numerator and denominator, subject to the condition that cos(t) ≠ 1. This simplification led us to the expression 1 / sin(t), which is the definition of the cosecant function, csc(t). Thus, we successfully transformed the left-hand side of the equation into the right-hand side, thereby establishing the identity.
This exercise demonstrates the power of combining algebraic techniques with trigonometric identities to simplify and manipulate complex expressions. The ability to express trigonometric functions in terms of sine and cosine, find common denominators, apply trigonometric identities, and cancel common factors are essential skills for anyone studying trigonometry and related fields. Furthermore, this problem highlights the importance of considering the domain of trigonometric functions and the restrictions that may arise due to division by zero or other mathematical singularities.
The identity -cot(t) + sin(t) / (1 - cos(t)) = csc(t) is a valuable tool in trigonometry. It can be used to simplify trigonometric expressions, solve trigonometric equations, and prove other trigonometric identities. By mastering the techniques used in establishing this identity, you will be well-equipped to tackle a wide range of trigonometric problems and deepen your understanding of this fundamental area of mathematics. This journey through establishing this trigonometric identity has not only reinforced our understanding of individual trigonometric functions but also illuminated the interconnectedness and elegance of trigonometric relationships as a whole.