Proof Of Trigonometric Identity (1 + 1/tan²A) (1 + 1/cot²A) = 1/cos²A - Cos²A

July 8, 2025 by Scholario Team 78 views

Prove the Trigonometric Identity: (1 + 1/tan²A) (1 + 1/cot²A) = 1/cos²A - cos²A

In this comprehensive guide, we will delve into the step-by-step proof of the trigonometric identity: (1 + 1/tan²A) (1 + 1/cot²A) = 1/cos²A - cos²A. Trigonometric identities form the bedrock of advanced mathematics, particularly in fields like calculus, physics, and engineering. Mastering these identities is crucial for problem-solving and a deeper understanding of mathematical concepts. This article aims to provide a clear, concise, and thorough explanation of the proof, making it accessible to students and enthusiasts alike. We will break down the complex equation into manageable parts, using fundamental trigonometric relationships and algebraic manipulations to arrive at the final result. Let's embark on this mathematical journey to solidify our grasp on trigonometric identities.

Understanding the Basics of Trigonometric Identities

Before we dive into the proof, it's essential to have a firm understanding of the fundamental trigonometric identities and their relationships. Trigonometry, at its core, is the study of the relationships between the angles and sides of triangles. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), each defined as ratios of sides in a right-angled triangle. Furthermore, we have reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot), which are the reciprocals of sine, cosine, and tangent, respectively. These functions are interconnected through various identities, which are equations that hold true for all values of the variables. Understanding these basic identities is crucial for simplifying complex trigonometric expressions and solving problems. For instance, the Pythagorean identity, sin²A + cos²A = 1, is a cornerstone of trigonometric manipulations. Similarly, the relationships tan A = sin A / cos A and cot A = cos A / sin A are frequently used to transform and simplify expressions. In our proof, we will leverage these fundamental identities to navigate through the equation and demonstrate its validity. Mastering these basics not only helps in proving identities but also enhances problem-solving skills in various mathematical contexts. By building a solid foundation in these core concepts, we pave the way for tackling more advanced topics and applications in trigonometry and beyond.

Step-by-Step Proof of the Identity

To prove the given trigonometric identity, (1 + 1/tan²A) (1 + 1/cot²A) = 1/cos²A - cos²A, we will start by manipulating the left-hand side (LHS) of the equation and gradually transform it to match the right-hand side (RHS). This approach involves using fundamental trigonometric identities and algebraic techniques. Our first step is to rewrite the terms 1/tan²A and 1/cot²A using their reciprocal relationships. Recall that cot A = 1/tan A, so 1/tan²A can be written as cot²A, and similarly, 1/cot²A can be written as tan²A. Substituting these into the LHS, we get (1 + cot²A) (1 + tan²A). Next, we will expand this expression by multiplying the two binomials. This gives us 1 + tan²A + cot²A + tan²A * cot²A. Now, remember the identity 1 + cot²A = csc²A and 1 + tan²A = sec²A. We can also express tan²A * cot²A as (sin²A/cos²A) * (cos²A/sin²A), which simplifies to 1. Substituting these into our expanded expression, we have sec²A * csc²A. To further simplify, we express sec²A and csc²A in terms of sine and cosine, using the identities sec A = 1/cos A and csc A = 1/sin A. This gives us (1/cos²A) * (1/sin²A). Now, we need to show that this expression is equivalent to the RHS, 1/cos²A - cos²A. To do this, we will multiply the terms and use the Pythagorean identity sin²A + cos²A = 1. Our current expression is 1/(cos²A * sin²A). We can rewrite sin²A as 1 - cos²A, substituting this gives us 1/(cos²A * (1 - cos²A)). The next step involves algebraic manipulation to show that this is equivalent to the RHS. This meticulous step-by-step approach ensures that each transformation is logically sound and based on established trigonometric principles, ultimately leading to the successful proof of the identity.

Detailed Breakdown of Each Step

Let's break down each step of the proof in detail to ensure a clear understanding of the process. Our goal is to transform the left-hand side (LHS) of the equation, (1 + 1/tan²A) (1 + 1/cot²A), into the right-hand side (RHS), which is 1/cos²A - cos²A. The first step involves rewriting the reciprocals of trigonometric functions. Recall that cotangent (cot) is the reciprocal of tangent (tan), meaning cot A = 1/tan A. Thus, 1/tan²A is equivalent to cot²A, and similarly, 1/cot²A is equivalent to tan²A. Substituting these into the LHS, we get (1 + cot²A) (1 + tan²A). This substitution is a fundamental step, allowing us to express the equation in terms of cotangent and tangent, which are easier to manipulate in subsequent steps. The next step is to expand the expression by multiplying the two binomials. Using the distributive property (also known as the FOIL method), we multiply each term in the first binomial by each term in the second binomial: (1 + cot²A) (1 + tan²A) = 1 * 1 + 1 * tan²A + cot²A * 1 + cot²A * tan²A, which simplifies to 1 + tan²A + cot²A + tan²A cot²A. This expansion is a crucial algebraic step that allows us to separate the terms and apply trigonometric identities individually. Now, we recall the Pythagorean identities: 1 + tan²A = sec²A and 1 + cot²A = csc²A. However, in this case, it's more strategic to keep the expression as is and focus on simplifying tan²A * cot²A. Since tan A = sin A / cos A and cot A = cos A / sin A, their product tan²A * cot²A can be written as (sin²A / cos²A) * (cos²A / sin²A). This simplifies to 1, as the sin²A and cos²A terms cancel out. Substituting this back into our expanded expression, we have 1 + tan²A + cot²A + 1, which can be rearranged as 2 + tan²A + cot²A. To proceed, we use the identities 1 + tan²A = sec²A and 1 + cot²A = csc²A again, but this time, we recognize that tan²A = sec²A - 1 and cot²A = csc²A - 1. Substituting these into our expression, we get 2 + (sec²A - 1) + (csc²A - 1), which simplifies to sec²A + csc²A. Now, we express sec²A and csc²A in terms of sine and cosine, using the identities sec A = 1/cos A and csc A = 1/sin A. This gives us sec²A + csc²A = (1/cos²A) + (1/sin²A). To combine these fractions, we find a common denominator, which is cos²A * sin²A. This gives us (sin²A + cos²A) / (cos²A * sin²A). We know from the Pythagorean identity that sin²A + cos²A = 1, so our expression simplifies to 1 / (cos²A * sin²A). The next step is to rewrite sin²A in terms of cos²A using the Pythagorean identity sin²A = 1 - cos²A. Substituting this into our expression, we have 1 / (cos²A * (1 - cos²A)). To show that this is equivalent to the RHS, 1/cos²A - cos²A, we need to manipulate the expression further. We start by finding a common denominator for the RHS, which is cos²A. This gives us (1 - cos⁴A) / cos²A. Our goal now is to show that 1 / (cos²A * (1 - cos²A)) is equivalent to (1 - cos⁴A) / cos²A. Multiplying both the numerator and the denominator of the LHS by (1 + cos²A), we get (1 + cos²A) / (cos²A * (1 - cos⁴A)). However, this does not directly lead to the RHS. Instead, let's revisit our expression 1 / (cos²A * sin²A) and rewrite it as 1 / cos²A * 1 / sin²A. We know that sin²A = 1 - cos²A, so we have 1 / cos²A * 1 / (1 - cos²A). To reach our target, 1/cos²A - cos²A, we need to combine the terms. Let's try a different approach. We start again from 1 / (cos²A * sin²A) and multiply the numerator and denominator by cos²A. This gives us cos²A / (cos⁴A * sin²A). Now, we rewrite sin²A as 1 - cos²A, resulting in cos²A / (cos⁴A * (1 - cos²A)). This still does not directly lead to the RHS. Let’s go back to 1 / (cos²A * sin²A) and use the Pythagorean identity sin²A = 1 - cos²A. We have 1 / (cos²A * (1 - cos²A)). The right-hand side (RHS) is 1/cos²A - cos²A. We can rewrite the RHS with a common denominator as (1 - cos⁴A) / cos²A. Now, we need to show that 1 / (cos²A * (1 - cos²A)) = (1 - cos⁴A) / cos²A. This step requires a clever manipulation. We can rewrite the RHS numerator as (1 - cos²A)(1 + cos²A), so the RHS becomes (1 - cos²A)(1 + cos²A) / cos²A. The LHS is 1 / (cos²A * (1 - cos²A)). To equate LHS and RHS, we multiply the LHS by (1 + cos²A) / (1 + cos²A), which gives us (1 + cos²A) / (cos²A * (1 - cos²A) * (1 + cos²A)). This simplifies to (1 + cos²A) / (cos²A * (1 - cos⁴A)). This also does not directly give us the result. We seem to have reached an impasse. Let's re-evaluate our approach. We have LHS = (1 + 1/tan²A)(1 + 1/cot²A) = (1 + cot²A)(1 + tan²A) = 1 + tan²A + cot²A + tan²A cot²A = 1 + tan²A + cot²A + 1 = 2 + tan²A + cot²A. And RHS = 1/cos²A - cos²A. We made an error in our initial understanding of the problem. The correct RHS should be 1/cos²A - cos²A, not 1/cos²A - cos⁴A. Let's correct this and proceed. Rewriting RHS with a common denominator, we get (1 - cos⁴A) / cos²A = (1 - cos²A)(1 + cos²A) / cos²A = sin²A(1 + cos²A) / cos²A. Now, let's simplify LHS again. LHS = 2 + tan²A + cot²A = 2 + sin²A/cos²A + cos²A/sin²A = (2sin²Acos²A + sin⁴A + cos⁴A) / (sin²Acos²A). We know that (sin²A + cos²A)² = sin⁴A + 2sin²Acos²A + cos⁴A = 1. So, sin⁴A + 2sin²Acos²A + cos⁴A = 1. Thus, sin⁴A + cos⁴A = 1 - 2sin²Acos²A. Now, LHS = (2sin²Acos²A + sin⁴A + cos⁴A) / (sin²Acos²A) = (1) / (sin²Acos²A) = 1 / ((1 - cos²A)cos²A). RHS = (1 - cos⁴A) / cos²A = (1 - cos²A)(1 + cos²A) / cos²A = sin²A(1 + cos²A) / cos²A. To prove LHS = RHS, we need to show that 1 / ((1 - cos²A)cos²A) = sin²A(1 + cos²A) / cos²A. This simplifies to 1 / ((1 - cos²A)cos²A) = (1 - cos²A)(1 + cos²A) / cos²A. Multiplying both sides by cos²A, we get 1 / (1 - cos²A) = (1 - cos²A)(1 + cos²A). This simplifies to 1 / sin²A = sin²A(1 + cos²A), or 1 = sin⁴A(1 + cos²A). This is not true in general. There seems to be an error in the identity itself or in our interpretation. Let’s verify with a simple value. Let A = 45 degrees. Then cos A = 1/√2, cos²A = 1/2. LHS = (1 + 1/tan²A)(1 + 1/cot²A) = (1 + 1)(1 + 1) = 4. RHS = 1/cos²A - cos²A = 1/(1/2) - 1/2 = 2 - 1/2 = 3/2. The LHS and RHS are not equal. Thus, there is likely an error in the original identity statement.

Common Mistakes to Avoid

When working with trigonometric identities, several common mistakes can hinder the process of proving them. Recognizing and avoiding these pitfalls is crucial for success. One frequent error is misapplying or misremembering fundamental trigonometric identities. For example, confusing the Pythagorean identities (sin²A + cos²A = 1, 1 + tan²A = sec²A, 1 + cot²A = csc²A) or incorrectly using reciprocal identities (such as sec A = 1/cos A) can lead to incorrect substitutions and derivations. It's essential to have a solid grasp of these core identities and their variations. Another common mistake is making algebraic errors during simplification. Trigonometric proofs often involve complex algebraic manipulations, such as expanding expressions, factoring, combining fractions, and simplifying terms. Careless mistakes in these steps can derail the entire proof. Always double-check each algebraic manipulation to ensure accuracy. A third pitfall is not strategically choosing which side of the equation to start with. In many cases, one side is more complex than the other and offers more avenues for simplification. Starting with the more complex side can often lead to a more straightforward proof. Additionally, failing to recognize opportunities for simplification is a common issue. Sometimes, a trigonometric expression can be simplified using a particular identity or algebraic technique, but the opportunity is missed. Developing pattern recognition skills and staying familiar with various simplification strategies can help overcome this. Another mistake is not expressing all terms in sine and cosine. When in doubt, converting all trigonometric functions to sine and cosine can often reveal hidden simplifications or lead to a clearer path towards the solution. This approach can help unify the expression and make it easier to manipulate. Lastly, giving up too early is a common mistake. Trigonometric proofs can sometimes be challenging and require multiple attempts and approaches. If one method doesn't work, don't hesitate to try a different strategy or revisit previous steps to identify potential errors. Persistence and a willingness to explore different avenues are key to success in proving trigonometric identities.

Conclusion and Key Takeaways

In conclusion, proving trigonometric identities requires a strong foundation in fundamental trigonometric relationships, careful algebraic manipulation, and strategic problem-solving skills. While our initial attempt to prove the identity (1 + 1/tan²A) (1 + 1/cot²A) = 1/cos²A - cos²A revealed a potential error in the original statement, the process highlighted several key takeaways. First, a thorough understanding of basic trigonometric identities, such as the Pythagorean identities and reciprocal relationships, is crucial. These identities serve as the building blocks for more complex proofs. Second, meticulous algebraic manipulation is essential. Errors in expansion, simplification, or substitution can derail the entire process. Third, strategic thinking is vital. Choosing the appropriate side to start with, recognizing opportunities for simplification, and converting terms to sine and cosine are all valuable strategies. Moreover, persistence and a willingness to re-evaluate the approach are key to overcoming challenges. If one method doesn't work, exploring alternative strategies or revisiting previous steps can often lead to a breakthrough. Finally, it's important to verify the identity with specific values to ensure its correctness. If the left-hand side and right-hand side do not match for a particular value, there may be an error in the identity statement itself. By mastering these skills and strategies, one can confidently tackle a wide range of trigonometric proofs and enhance their mathematical proficiency. Trigonometric identities are not just abstract equations; they are powerful tools with applications in various fields, including physics, engineering, and computer science. Therefore, a solid understanding of these concepts is invaluable for students and professionals alike.